Bangkok is much poorer than Taipei and much more touristy. Lots of Europeans and Australians visit Thailand, so the locals are used to seeing Western faces. In terms of "ordinary" tourist attractions (temples, markets, etc.) Bangkok has enough for a pleasant 3 or 4 day stay. And of course you could spend months in other parts of Thailand.
Taipei is not a very pretty city, but it has one of the best museums I have ever been to (The National Palace Museum) which has an incredible selection of Chinese art. For the rest it is a relatively affluent city with the usual big shops, cinemas, etc.

For gay life, Bangkok is great for sugar-daddies and rent-boys, but the situation for others is improving and in places like DJ Station (discotheque) and Babylon (sauna) it is possible to meet local guys who are not looking for money.
Taipei is great if you like Asian (Chinese) men. There are relatively few Westerners, so it is easy to meet local guys who like Caucasians. Most speak English well. The night life is pretty wild in some discotheques, with bare chested guys openly using all kinds of pills and powders that you wouldn't expect people in Taiwan to use.........