I have a reason why this type of set up using the ATV with cable is good.
The ATV's UI is so much better than the cable companies. The cable companies' STBs are large, slow, buggy, and have outdated UI. Plus, many companies charge for HD. If you have a few HD STBs that you are renting, the costs add up quick. In a years time, you could buy a couple ATVs.
I pay for a cable tv sub. It only costs a little more than internet alone, and it includes Showtime for 12 months.
I do not use Comcast's crappy STB. It has never been hooked up. It is sitting in the box that it was shipped to my house and I have never taken it out. It was the free one that came with my sub, so there is no monthly fees for it.
I use ATVs all over my home. If there is a station that I am paying for, but it not available on my ATVs I do not watch it.
For the reasons listed above. I will reverse the question on you:
Why would you watch something on the DIRECTV box if you can get the same content on an Apple TV?
I've considered doing the same. The price for cable tv is not much more than the internet that carries it. Some offers even pay you to take cable! But unfortunately, so many of the AppleTV apps that work on a cable subscription aren't available in Canada. I'd return to buying cable TV after 9 years (first AppleTV!) if I could download all the individual channel apps and watch them that way.