I guess my primary enjoyment comes from the awesome feeling of seeing myself improve over time. This was much easier to see when I was still working my way up from Normal difficulty, and now I've kind of plateaued in the sense that I don't really have a good measuring stick for myself anymore. Also I guess I'm a natural show-off when it comes to things I feel I'm good at, and I know that when I first saw the game, I was basically in awe in terms of how people could play it. I think I rather enjoy passing on that feeling to someone else, especially when I get to show them a replay of myself playing at Normal as well. Embarrassing, but still awesome.
That's my general motivation for playing games, but for Touhou specifically, I can sum it up to the absolutely wonderful overall presentation. Music, design, lore (I'm still in awe at some connections that ZUN explains in his side books), fun-level, etc. I can't think of any particular area where the games are sorely lacking (ok the art, but the design is still great), and seeing the colors and hearing the music every time has a nice sense of familiarity that I just enjoy because I know that I'm sure as hell going to love what's coming. I guess I also like the somewhat relaxing part of playing the game. Sure I might rage every once in a while, but let me tell you that the community around competitive multiplayer games like LoL and Starcraft get rather grating after a while, and Touhou is a nice way to just kick back and enjoy a game just for what it is, and not necessarily because of some socially imposed import. And sometimes I get that feeling of social constructs and whatnot being far too stressful and rather fake, so like I said before, it's nice to just "enjoy a game just for what it is, and not necessarily because of some socially imposed import."