I play trying to get full combos and perfects and after that i try to use mods. It's satisfying to see my score getting higher each time in the rankings of each song thinking that I did better than many other players, but if I barely pass one and see myself at the bottom of the ladder I get demotivated a bit.
I play osu for the FUN competition. I personally don't think osu would be nearly as fun if it didn't have leader boards or any sort of competition with others. The fun and competition play into each others hands.
I do play osu!mania [4K] somewhat competitively -- I will sit down and grind for better ranks if I know that I can improve ranks. I am planning on getting a tablet for playing osu! standard, but I can't see myself extremely competitive with that aspect of osu! (at least, not yet).
I mostly play competitively, since rhythm games are the only games I consider myself pretty good in. I do sometimes play for fun, but that's rare. And even then, I often get frustrated if I don't score well.
Most of my friends are in the top 50 of Ireland (trust me that's not hard to do) but we have great fun fighting for ranks in songs and ranks in the Irish ranking system. I have had a few frustrating moments repeating the same song again and again just to best my friends score but most of the time we play multy while on skype and all have a good laugh. I assume when its no longer fun I'll stop playing ^_^
Well, competitively actually owo I have a few friends that play Osu! as well and we're trying to see who has the highest rank / pp / S and SS's / etc So uh... yeah.
Fun of course. Improvement is fun, getting high scores is fun. Finding new patterns in maps is fun. Seeing your effort and concentration pay off is fun.
But above all, I just like rythmically clicking circles