Consumer Action Line in my local newspaper just ran a comparison between these two 'black' boxes. The TIVO won hands down. There have been many reported problems with the HR20 - everything from flat out failures to missed recordings. Get the TIVO.
There are 10 types of people in the world. Thos that understand binary and those that don't.
The HR20 is a complicated new box and the cornerstone of D* HD strategy. Neither the TiVo S3 or the HR20 are perfect yet. Look at the wider picture; consider total cost over a couple of years and look what HD programming is promised by the service operator. With both options you could be committing yourself to a 2-3 year contract.
Also consider a monthly lease from your cable company of their own HD DVR if they offer one. If they do not tie you into a long service contract then you can postpone your decision for a few months until the S3 and HR20 falls in price and any issues have been worked out.
I think that the Tivo Series 3 is not compatible with D*V, can get the HR20 for 299 plus they will put new dish 5 LNb up for free, already have hd premier package so no monthly fee for dvr use. 1 yr commit. Is the Tivo 3 Series m-peg 2 ?. I no the HR20 is m-peg 4.