The quick and dirty Dervs take:

-Free is a section mainly populated by novice your younger roleplayers who feel like it's a good starting point, or just like the easy going flow and lack of structure. I actually stopped roleplaying for years because my original forum's singular roleplay forum shifted into a series of short and fast posts (we're talking dozens a day and if you weren't online, you were left behind), and it's definitely not my speed or style. It's great if you want that kind of thing and something that's not far off from a chatroom style RP, but I feel like speed posting is the worst thing ever. It's a huge time commitment and it pressures you to not take any days off.

-Casual: The great generalist section that exists on a spectrum of people who should be writing in Free but fear the stigma of it to people who should be writing in Advanced but either think they aren't good enough writers for some reason or the pace is too slow (fair point there). In my mind, Casual should be the relaxed middle ground forum where players set up proper OOCs, have some standards that aren't usually detailed, and people write 2-4 regular sentences on average. I like the idea of that, but my experience with Casual's usually  been that games have a hard time getting going, lax standards mean you end up with way more problem players or incompatible writing styles (e.g. opposite ends of the Free-Advanced spectrum), and a hard time retaining player interest. Not saying that's the norm, just personal experience. It's actually why I started writing in Advanced.

-Advanced: For people who want to make sure that most players are on a similar level of writing skill and view roleplaying more as collaborative writing instead of games. Higher standards for applying and usually a closer community due to the lower number of players, it's great for those of us who love long and detailed posts that evoke reading a novel or just some really insightful looks into the characters and descriptive imagery. While individual posts take longer to write, the posting rate is usually lower than other sections so if you have a busy life and can't post every single day, working on a parageaph or two every few days is a good way to go. It's not about being pretentious filler, like [@Tex] suggests, it's like a different style for people who want detailed posts that are more about the story than the game mechanics and actions. The argument that you can turn 8 paragraphs into 2 is missing the point entirely; just because it doesn't appeal to you doesn't mean it's not what other people want. Personally, I find short posts that are pretty much jut functional parts without much description or character insight kind of lifeless, but again, different tastes. I don't judge Free players for loving the kind of roleplaying that nearly killed my love for it years ago; they just want different things than I do. It's subjective, like music genres.

1x1: I love the idea of having a singular partner, I just never see anything that appeals to me. A lot of generic pairings, plot ideas that don't click for me/ seem a bit too catered for the thread poster, or forced canon characters just don't really click for me. If I do 1x1s, it's with people I've been writing with for a while for the most part, although that's not to say I don't look for new parters. Just the nature of most interest checks don't fit what I'm into.

Nation: No real opionions on this one since I haven't participated, I just worry that it becomes a real number cruncher and people might be in it to win it rather than just play a game for the sake of fun.

Arena: Again, no experience on the site. I just figure there's a sort of community rules and guidelines for who wins a fight, I'm not sure how I feel about profiles tracking win/lose ratios.

Tabletop: Never tried it, I know people were all but demanding it.