CatBus said:

I can't even think of an actual problem I've really encountered that it solves.

Our HTPC allows us to:

1. Watch things we download from Usenet on our TV; e.g., a 1.5GiB episode of Modern Family downloads in less than 5 minutes

2. Watch "webisodes"; e.g., Cinemassacre.com videos

3. Retrogaming (mostly NES using the Nestopia emulator) – we have USB NES controllers, so it mostly feels "real"

4. "Modern"gaming; e.g., we're playing the AVGN game on Steam right now

We don't use our HTPC exclusively, of course. We have tons of equipment, physical media, and video game consoles. Our HTPC does, however, act as a really convenient supplement.