Unfortunately Dan's answer is both wrong technically and in reality.
I'll say that DirecTV's picture is often times better. The quality of a picture has nothing to do with the bandwidth available in a specific media; i.e. radio frequency signal versus fiber optic light signal. Verizon's fiber to the home uses one single frequency of light to deliver data to the receiving device. A satellite uses a range of frequencies allocated to it. Raw bandwidth from a satellite to the home could easily be higher than the equivalent amount of bandwidth available from the Verizon fiber but whether it is or not really doesn't matter.
The reason why none of that matters is because what dictates the quality of the picture is the original picture format, the transmitted/received format, the compression algorithm your provider uses for the image and audio data, and how much compression they choose to use based on their desired bandwidth consumed per channel. Both providers are sending the content of every channel to every subscriber at all times since both types of devices are passive receivers of the television data. Content on demand (pay per view, on demand, etc.) adds to bandwidth requirements. Internet service (which fios provides and DirecTV does not) add to the bandwidth requirements.
I've used both services and I've found that typically DirecTV seems to have slightly less compression in their high def signals than Fios does. Additionally, keep in mind that Fios does NOT broadcast anything in 1080p format; while DirecTV does have some content in 1080p/24. This can be a huge difference if you're watching movies since DirecTV will give you the version you'd see at a movie theater or from BluRay and Verizon will give you something else that somewhat resembles what you should have seen.
So movies aside, you can see the difference between Fios and DirecTV if you're watching high def sporting events (with fast motion) or movies that contain scenes with either a very large number of colors or a lot of dark, but distinct, colors. In either of those situations, you can often see more artifacts on the screen on Fios; i.e. pixelation around edges, blotches in the dark areas, etc. I'm not up to speed on the fancy lingo for how you properly describe these phenomena but you'll see if it if experience both services. My guess is Verizon is just limited in bandwidth from head end to household due to bundling internet service into the same distribution network but who knows.
Downside to DirecTV is that contrary to what they say, when it rains hard or there's heavy cloud cover, there will be interruptions. I'm in South Florida and see it regularly. It's annoying.
Also, there's an issue of pricing. If you just have one TV, AND you use high speed internet, you may find that Verizon is the better deal since Verizon will charge you a lot more for internet-only service versus what they charge overall for Fios internet+tv. I used to use Fios only because I had high speed internet, which is very very good on Fios, one single TV and the television service. If I had gone with DirecTV with a similar package, and dropped tv from fios, the overall cost would have been higher.
Fast forward, bigger house, more tv's, then it made sense to go with DirecTV. Verizon charges a lot for DVR equipment while DirecTV's Genie lets me record up to four shows at once, and I have auxiliary devices on two other tv's so with one dish I can be watching three shows on three tv's live or from DVR, the picture quality is better since DirecTV compresses less, and I still have the Fios internet service. I save about $20/mo overall versus what I'd have had to pay for Verizon's DVR charges, with more limitations, lower picture quality and less recording time unless I get the DVR that lets you connect an eSATA external hard drive to it which I'd also have to purchase.