I visit BoardGameGeek quite frequently, as I play Euro games quite often - much more often than Axis and Allies which is now only confined to PBEM since I haven't found people interested in playing it offline.
Likes: Diversity of games, rating system, diversity and mass of community.
Dislikes: Clunky interface, as has been mentioned. Very geeky feel, which of course is a given when you consider their name.
In response to the interface dislikes that others share, FYI I hardly spend any time on the front page of their site. For the longest time I could never make head or tail of it. What I've always done is do a search for a particular game I like and start from there, browsing the forums for that game, and sometimes following links to other games. In order to find a new game I might sometimes scroll through the top 100 or top 200 rated games on the site, but it's rare that I'll find a new game that way - most often I've heard of the game before I look it up on BGG.
I also see no point for all the badges and gold, but I figure it's the equivalent of things like forum avatars, post counts, and custom titles here, which I equally don't care about (I have avatars disabled, and would disable titles if I could). When I ran a forum in the past, I would ban people who would post about their post count. Guess quite a few people here would be gone, then