Hi there Kiku!
First let me say that people who enjoy playing anthro characters are not a collective. I can't give you a set of hard and fast rules telling you all the do's and don'ts to playing furries, because everyone has their own tastes, and what works for me may not be for someone else. I can only tell you what I like, and what playing an anthromorph entails to me. I've been roleplaying for a ridiculously long time, and primarily preferred playing anthro characters for the past... say 6-10 years, so I have a fair bit of experience under my belt. In addition to the things I like, I'll try and give a general overview of what I've seen, and prepare you for what you might encounter playing with other furs, but this is by no means a complete, comprehensive, all-encompassing list.
It's hard for me to pin down exactly why I like to play furs. If I had to take a shot in the dark, I'd say it started with the art. Drawn image references have always been much more appealing to me than photographic face-claims. I'm not sure I would have ever jumped onto that bandwagon if I'd been thinking about a photo-realistic dog-face superimposed onto a human body; but the right artist can make almost anything look appealing, and there are a ton of talented furry draw-friends out there, who like to produce wonderfully salacious pictures and animations that just look enticing. If you want to jump into that world, e621.net and furrybooru are my go-to sites for anthro porn, and you can find some great (and terrible!) work there. Just be advised that both have a lot of NSFW stuff, obviously.
You're correct in saying that a lot of people don't seem to reference the 'animal traits' as they play. My speculation is that these are people who are either new to playing furs, or those whose primary interest is that they find them visually appealing. They aren't 'doing it wrong,' that is a perfectly acceptable way to enjoy the subject matter, and if the only thing you're interested in regarding furs is the way they look, then that is nothing to feel ashamed of. Don't let anyone gate-keep you by trying to tell you that you should be doing more with the animal aspects of your anthromorphs!
Animal behavior is part of it for me. That's a fairly common kink in the real world, a lot of people enjoy roleplay as a kinky kitty, or a loyal dog that loves to please. To that end, I don't think playing an anthro would be that much different than playing a kemonomimi. You're totally in the clear to play a full fur in the same way you'd play one of your nekomimi characters.
That doesn't necessarily mean that furs are all looking for pet-play. Far from it, many don't like that sort of thing; however the freedom to act a little fluffy and silly, like a feral/quadrupedal animal might, can be extremely gratifying. I often try to think about a big, dumb, affectionate, over-sized dog when I'm playing my preferred wolf anthro. He's very excitable, in a way that might mirror or rhyme with a big doggo getting amped up when he hears his owner coming home from work. He's a bit clumsy, like he doesn't quite realize how big he is, but is extremely sweet and cuddly. He's loving and loyal to a fault, and embodies pretty much every other trait I've adored about pets I've owned. He tends to be extremely hard not to love~! As much as how he looks and acts might mirror an animal, however, I'm just not that into someone say... making him eat from a bowl on the ground, or putting him on a leash and having him go do his business on the lawn, rather than just letting him use the bathroom. There are definitely furs who do like that kind of thing, but it isn't necessarily typical either.
Kemonomimis are often a compromise, a sort of olive branch that gets extended to me, by someone who just doesn't find furries attractive, but still enjoy my writing and want to try out a story with me. Their distinct physiology makes for a few obvious differences between humans or mimis; furs have muzzles, which can be awkward to think and write about for some. (If I have a long, lupine snout, it's gonna be a bit harder for you to sit on my face than if it were more flat and human-shaped, for example. In contrast, a muzzle might make it easier to suck someone off, though depending on length it'd be a lot harder to get to your throat.) Muzzles and feet/paws are the main point of difference there... aside from genitals, but I'll work up to that.
Almost all furs have muzzles where applicable, that's part of what really sells the aesthetic. Some choose to play with totally human-shaped feet. These are called plantigrade, because you carry your weight on the plantar region/heel. Some opt for a more animal-like appearance to their legs, carrying their weight on their toes, the same way a cat or dog would (digitigrade). The former may still have a smooth, human-like bottom, or they may have more animalistic 'paw-pads,' soft, smooth, rounded regions typical of what you'd find on the bottom of your kitty's foot. Personally I think paws are cute, pads are soft, and I love to imagine how they'd look and feel, so I typically play with digitgrade pads, but again, there are no wrong answers here.
Many of the anatomical differences are internal. For example, canines have a sense of smell that is many times greater than a human's. The difference in anatomy that makes that happen (apart from muzzle-shape), would be in the olfactory bulbs of the brain, so that's not something you have to worry about in the description. Purring is another you may have done as a neko; I don't know enough about feline anatomy to tell you exactly why they pur, but it happens beneath the surface, so you're totally within rights to make YC's throat rumble when they're happy. (Or scared or hurt, cat's pur for all sorts of reasons!) These kinds of traits can be exactly the same between a -mimi and a fur, it just depends on whether that's something you're interested in playing with.
As I've experimented with different aspects of roleplaying a humanoid animal, I've found an amplified sense of smell to be something I greatly enjoy writing. Writing senses (mostly smell and hearing, maybe night vision) that I don't have is a great creative exorcise, that helps my brain think outside the box and generate content people don't see all over the place. I try to use similes, metaphors, and generally poetic language there, since describing a smell you can't smell would probably be as difficult as trying to explain a color you can't see. How would you explain red to a blind person? Most would say fiery hot, angry, loud, etc. I don't always want to hyper-focus on all of the various, conflicting smells going on around my character all of the time, however, so I often write my furs with the natural, unconscious ability to 'tune it out,' with their super-human senses only coming into play when they concentrate and focus on them.
If you want an example of that in media, there's a great one in the anime Beastars. There's a scene where the male lead is frantically searching for the female lead. He stops, takes a deep breath through his nose, and you can see the mist of her scent being slowly sucked into his skull as he tries to identify it and separate it from the crowd. Wish I had a GIF to share, I'm surprised that I haven't been able to find one. Being able to tune it out like that would probably be a great option for you, since you can really get into that if you want, or just totally ignore it and play YC as a human if you don't.
Pheromones are tied into that sense of smell. Because their noses are so much more adept than ours, a lot of communication in the animal world is olfactory. They may scent a favorite object, rubbing their aroma against it. They may scent a partner, caressing them with their erogenous zones, or scent glands in their cheeks or throat, to rub off on them, and sort of 'mark their territory' to tell other furs that might look at them as a mate to stay away.
Aside from those super senses, there's also a ton of animal behaviors that furs like to play with. Heat/rut is one of the most popular there, which you may have encountered as a -mimi. There's essentially as many variations on exactly how to play that as there are furs. It's all based on the concept that animals have different breeding seasons, and usually takes the form of extreme arousal and an (often inconvenient) instinct to mate. Think about it like a character, when she enters her heat, having an intense, natural aphrodisiac pumping through her bloodstream constantly. Sometimes all furs are the same, with them having generalized heat. Some writers choose to make their cycles more closely based on the animals they're playing. Others still totally make it up to suit their tastes, or ignore that aspect altogether.
Other instinctive behaviors are fairly common too. Again, Beastars is a great watch, and a wonderful way to dip your toes into the concept of an instinct that an anthro's rational mind might have to wrestle with. Many feel overly protective over their mates. Some may feel an irresistible urge to chase a prey animal when they see them run. Everyone might tense up in fear and apprehension, getting ready to fight or flee, when a careless wolf yawns and reveals his razor sharp teeth, a sign of aggression in the animal kingdom.
Speaking of, posturing can be important to some. Ears flattening against your head can mean you're embarassed; drawing back and away could mean you're scared or angry. Same with a waggle or bristle of your tail. Many people don't go into detail on that at all, but if you do some cursory research on animal body language, it can help get how your character is feeling across by showing and not telling; which I adore for feeling a little less on the nose. I don't always utilize it myself, however, and you may have experience with that as a nekomimi.
There can also be a sort of natural heirarchy, based on the smell of a fur and/or their natural proclivity to dominate or defer to a more dominant beast. Alpha/beta/omegaverse is a genre I'm new to, that claims to be based on the dynamics of wolf packs, where you have dominant alphas in charge of everything, worker-bee betas, and submissive omegas. A/B/O isn't, wolves don't actually have social organizations like that, (though some have pack omegas, and other animals have similar hierarchies); and that entire idea is misinformation being spread, based on early misconceptions about wolf biology. It's still a fairly interesting setting to write in though!
To be clear, ABO often doesn't mix with the anthromorph fandoms. Despite having many similar themes, ABO writers often prefer to play humans, and I've talked to a bunch that think furries are gross. xD
Knowing a bit about animals in general, and the specific species you'll be wanting to play and play with can be useful; but it isn't necessarily, because a lot of people make it up on the fly, choosing to act how they feel an 'animal' would, in an amalgam of all of these different types of behaviors, rather than basing it heavily in reality, and modeling it after a specific species.
Breeding can be a big thing for some. There are furs out there who love hybrids, and those that detest them, and play with in-universe rules that only the same species can produce viable offspring; so a wolf wouldn't be able to have a baby with a horse, regardless of whether they're in heat.
My take on that is that 'anthro' is the species, unlike all the feral animals they look like. A sheep and a bear-morph would both have an identical genetic code, but their different appearance (different phenotypes) is a result of different genes within their genome being expressed. I'm not a fan of hybrids, so I typically make them rare, and have the offspring of mixed-species(/phenotype) parents take after one or the other. e.g. a wolf and horse sire a wolf or a horse daughter, not a hybrid (in most cases).
That is definitely not typical, it's my own little head-cannon, but should give you an example of one of the many ways you can handle that kind of subject matter.
Circling back to anatomy, animal genitals are also somewhat popular. In my experience, the main interest there involves canine and equine penises. Equines are so popular, I assume because of their famous size ('hung like a horse' is a saying for a reason, I suppose), that a lot of furs impose them over other species. Bulls, deer, elk, moose, etc. are often drawn with big ol' horse dicks, despite having their own anatomy in reality. Canines seem to be a close second in popularity, because they have a knot at the base, a thick ball that inflates inside of their partner during orgasm to lock them together, and keep his cum inside of her. There's plenty more, sharks and some reptiles have two dicks; canines have barbs on the tip of theirs, which can be painful and stimulate ovulation in their partners; something that is pretty divisive among those that play them.
In reality when a dog or wolf mates, after mounting they insert their semi-erect penis into the bitch, which gets fully erect inside of her vagina. When they orgasm, the bulbous knot at their base inflates, and special muscles inside of her canal grips it to tie them together for anywhere from several minutes to an hour. The vast majority of furs don't play their anthro canines like that, allowing them to achieve full erection, and slipping the knot in at the very end, but many still like the kink of being tied together. So again, none of this is really grounded in hard reality, it's pretty much a bunch of creative, kinky people taking the most appealing bits of animals to write with, and leaving the rest on the table. If someone tries to tell you you aren't doing it right, they probably don't even know what they're talking about.
Cool animal dicks and the interesting things they can do is part of the appeal to me. Trying one out after a few years of writing humans ruined me. Human dicks are boring now, compared to a horse's fat flare or a dog's hard, hefty knot. Animal cocks are a popular interest, but there are many furs who don't like them, and prefer to use human genitals. Those that don't may be willing to play human anatomy anyway, if the thought of ERP with a sheathed, red canine penis makes you uncomfortable and squicks you out. Then again, I don't have much experience there, but there's usually no harm in asking if you aren't sure. You occasionally see animal vulvas being used in anthro circles, but in my experience those are extremely rare, and human vaginas on furs is pretty much the expected norm. Just like dicks, if they are present, they're usually on a canine or a equine, with other vulval variations being exceptionally rare. Multiple breasts aren't necessarily tied into them. Some play purely human boobage. Some like a 'main set', with either smaller breasts or rows of nipples underneath. Some go hog-wild, and have six pairs of boobs. Two human-like tits is the most popular configureation by far, based on what I see at least, but people like all sorts of stuff there.
Anonymity may also be part of the general draw for some furs. Acting as an armchair psychologist, I've wondered if part of the appeal would be like wearing an animal mask to the nth degree. Instead of doing something naughty and kinky with another human, it may just feel safer and more relaxed to play with something that's a bit different, but still attractive, and may reduce feelings of transference between the OOC and IC, in theory at least.
I hope some of that was useful! I probably could have organized it a little better. If you'd like me to clarify anything, or have any more specific questions, I'd be happy to consult with you and do my best to give you answers. Above all, just don't get too hung up on anything, and have fun with it, and I'm sure you'll do fine!
As a final word of warning... furs can be... a bit... weird and unstable. xD Some have a fair bit of emotional baggage and get really into the concept of 'being' their 'fursona'. Some are really into absurd hyper kinks, or vore, or filth, or other things a lot of people find bizarre and off-putting (if being a fur didn't fall into that category enough on its own), and they can be loud and obnoxious about it. There are others out there who are laid back, and just like it for the fun of it, so try not to get discouraged, or lump us all together; some of us are pretty sane, relatively speaking! x3 Not gonna lie, if you start looking for other furs to play with you will have some bad experiences, but hopefully you'll have some great ones too... and I don't think that's all that different from non-anthromorph-related ERP!
Have fun fleshing out your new furry character!
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