I'm pretty familiar with the inner workings of tivo; I've been hacking them since they came out.
What they do thats primarily different from the directv box is that the directv box tries to do an awful lot of things in real time like guide data processing, record/stop record decisions, etc. All too often they end up with too much to do and the user hits a remote control button and nothing happens for 3-5 seconds.
Tivo processes its guide data all at once, often in the middle of the night, and makes its to-do list periodically rather than in real time. While that sometimes leads to things like the much maligned "please wait..." when changing the order of the season pass list, it doesnt make the box have to do heavy duty work while you're just recording and watching shows.
So two different approaches: one built an architecture that doesnt overburden the box, the other just kept throwing wood on the fire until they needed a new box to handle the load.
But directv wants to showcase a lot of their programming and offer a lot of real time stuff to their customers and the standalone tivo really doesnt bother with any of that. Frankly I never use most of the goofy bells and whistles. I just want it to reliably record shows, play them back, and respond to the remote when I press a button.
My five year old cracks me up. He'll press a button on the directv remote and nothing will happen for a few seconds and he'll yell "HEY! HELLO?!? I'm pushing a button over here!".