I've never used a Tivo, but I do use BTV. I can see the advantage of Tivo Series3 being:
1. Plug it in and it works.
2. Digital CableCard Support - get all cable channels, not just the unencrypted ones.
3. Not as much maintenance and "tweaking" required
Advantages to BTV (or other HTPC's)
1. You control the content
2. You can watch from any room if you use BTV Link, or stream over the internet etc.
3. More flexible ( for example, check out fonceurs BTV Negociator for some of the things you can customize in BTV)
4. Ad-detection and skipping
5. Squeeze (compress) shows to take less space on the hard drive
6. I'm probably forgetting something - I know there are other advantages here....
For a story about a person who moved from an HTPC to a Tivo Series3,
read this story. It's a good chronicle of what was good and bad about the move.