I don't actually have a S3 unit and so I can't answer any of these questions from actual ownership experience. However, comparing a TiVo to a 3Ghz P4 isn't a fair comparison. The TiVo is a specialized computer that is designed specifically for video playback. It has dedicated MPEG decoding hardware that the typical PC lacks and I'd hazard a guess the Linux OS it runs has been tweaked to provide better support for real-time operations. Most desktop OSes just aren't geared for that. I don't believe the S3 handles 1080p at all, since that's not a broadcast standard. The specs for it only say "standard modes including 1080i & 720p widescreen". If it works at all, it will probably work just fine (for the reasons mentioned previously).
Now, I realize I haven't technically answered your question, but maybe this is helpful somehow.