For a long time, the DirecTV DVR was a Tivo DVR.
However, DirecTV has decided to branch out on their own and build their own Satellite receiver with DVR built in. I think the rationale is that they don't want to share money for the service with a third party and/or have another brand competing with their own brand.
Their advertising is a bit odd because DirecTV claims "NOW" you can record, pause, and rewind live TiVo. What makes this odd is that you've been able to do it for over 5 years with the DirecTV TiVo so I'm not sure where the "NOW" bit comes in.
From what I've heard, people aren't as happy with the new DirecTV DVR that isn't a TiVo compared to the TiVo. I don't have the satellite DVR, I have used Cable DVRs and can attest to the fact that they have fewer features and are harder to use than TiVo.
DirecTV's contract with TiVo in 2007 and so there is the possibility that DirecTV will try to make a go of it with only their own DVR and without TiVo after that time.
Another thing to consider is that DirecTV has launched some satellites with the intent of using MPEG4 rather than MPEG2 as the video standard so they can get more digital channels and HD channels into the same bandwidth as MPEG2. The current DirecTV TiVos only support MPEG2 and will not likely support the additional channels. I've read about the HR20 HD MPEG4 DVR being out this year 2006.
So, given that, if you want DirecTV and want a DVR, it pains me to say it being a TiVo fan, but you are probably in a better situation getting the DirecTV DVR instead of the TiVo.
If you are Cable subscriber and want HD, I'd suggest just leasing the box from the cable company until the TiVo Series 3 box comes out later this year 2006 (summer or fall is my guess). That's what I'm doing until then.