I'll fill you in on what I understand, being that we just finished installing the DirecTV DVR today!!
I didn't have a ton of time to research, as my parents gave us some $$ towards it for our early anniversary present, and the deal Circuit City had on it expired yesterday!
However, the only thing I could tell as the real difference between the DVR done by DirecTV and the one that is purely Tivo were some of the more advanced features - TivoToGo (which I think was something like being able to download shows straight to a computer to burn to DVDs), the ability to tell your Tivo to record anything from anywhere with an internet connection (which I think is pretty cool!

), and being able to show your pictures/play music stored on your computer through your TV. There is a chart on Tivo's website:
http://www.tivo.com/1.0.chart.asp
But, I do believe all the normal functions of Tivo are there on the DirecTV DVR - season pass features, the search features, etc. (At least - I hope so! There's no turning back now!

)
I think the DirecTV one is somehow linked with Tivo - it no longer says it on the box, etc., like I think it used to, but it still appears in the fine print on both their sites.
We were actually able to get the DVR R15 for about $99 with a $100 MIR - so, essentially free! It is supposed to hold upwards of 100 hours (although I understand that is dependent on the quality of recording you select). This was the best deal we found - even on the DirecTV website, they don't have as good of one for current subscribers. (Although I do believe they do if you're a new customer.)
We are recording our first thing tomorrow morning (hey - the Cotton Bowl, Denise!), as a trial run before we entrust it with the Rose Bowl while we're out of town!
We're excited! (And I hope this helps you - sorry for the length!)