The TivoHD should be able to record regular analog cable like your S1, with the added ability to tune ATSC antenna Hi-Def channels or use CableCARDS if you want to go that route. With broadband access, the newer features are kinda nice, like Yahoo Weather, Traffic updates, Amazon Unbox (when it works), and Swivel Search.
The only couple of drawbacks I could see are:
(1) the lack of video-inputs on the TivoHD, and no IR blaster. Meaning that you can't hook it into a satellite receiver or other setup if you ever decide to leave the cable company.
(2) The "transfer" offer involves enrolling your old TiVo unit into a recurring billing plan, so you have to be darn sure to call in and specifically cancel your old box before the "free year" is up.
If your S1 is getting that old, it might be time to put it out to pasture.