In my experience, having written on both sites, BMR is more casual based and faster. We don't really have an application process here, we have us in the welcoming committee and often other members of the site saying hi when you make your introduction and the low bar of making three posts somewhere on the site and waiting 24 hours to become a full fledged member. It's also more alt friendly, you can't have up to 5 accounts anymore but people do have the option of more than one if they choose to. BMR still has strict rules that are well enforced by staff, but is at least a little easier to navigate in that regard too. It's perfect for newcomers to the hobby of writing and experienced writers alike and because of that I think it has more engagement, but I haven't been on Elliquiy for a while so I'm not certain of that.
Elliquiy is much stricter in it's application process, a process that took a few days when I joined a long, long time ago but was in my opinion a very fun process. I'm not sure if it's the same now but in order to be given the keys to the castle you had to answer questions asked by staff and other members, some relevant to writing, some just fun and nonsensical, answer a pretty large list of questions and wait several days with only the application forum to keep you company. It likely exposed any red flags and behaviour that suggested the applicant was seeking to just "get off" and protected it's members from those likely to harass. The forum itself once you're past that, I remember it being very bright, harder to navigate but larger and being way more story and plot focused than what we have here, which I think has a nice and healthy balance of both plot based and smut based roleplays. I don't think Elliquiy is in any real way elitist, I just think they are cautious who they let inside. I'm hardly elite and I got in just by having fun with the application process.
I like it here better, I've had a more fun and social, relaxed experience here but I never had a bad experience there either. I do remember there was a pretty cool market available where you could buy and sell things like art and advertise your services, I'm not sure if that's still there or not since I'm not even sure if it has been more or less than ten years since I was active on the site. I think it's been more. The inclusion of a built in kink list that also had all kinds of information about settings somebody enjoys writing, where they liked to post, post lengths and expected activity/frequency, the "ons and offs" that I remember is a definite highlight I'd enjoy seeing here but it's not very hard to just write your kinks down in a post somewhere and point to it so it's not a big deal. I think those are the only things I really think BMR would benefit from. I like the default colour scheme better here, I think it's easier to find a partner here and it's a great place that caters for writers of every level. You can find a roleplay that consists of a single sentence per post if you like, or one that has 3000 words per post and have a lot of fun doing either. You can have a smut fest or something with little to no smut and you fit in just fine.
Elliquiy does send me a message on my birthday, which I do like. It's a nice touch.
Hope that helps highlight a few of the differences. Most of what I typed seems like babbling to me, but I only recently woke up. There is probably at least something relevant in it, if not then I'm so sorry!