I had Verizon for 4+ years, AT&T for the last 4+ years and 2 years of that overlapped where I had both.
I hardly ever had a dropped call with Verizon, but with AT&T I do have call issues like in the middle of a call they can't hear me but I can still hear them. I am now on my 6th iPhone (just got a replacement today) mostly for call issues.
Honestly I still have USB data cards on both networks and in areas where 3G and EVDO is deployed for BOTH networks there is hardly a difference. Before AT&T deployed 3G in my area, Verizon won hands down.
To me, the REAL difference is inside a building. With Verizon I swear I had great service in my basement, in the elevators of my university buildings, all with a strong signal and no call dropping. AT&T is a whole different story. I typically drop to EDGE or no signal in a building for my data card and even in my apartment sitting by a window when I am on the first floor. I also drop calls in elevators with AT&T.
Now that I have said my $0.02, it is subjective. It is all about where you live. I switched to AT&T because I have a family plan with 4 people and my mom REFUSED to get Verizon. Now because of my iPhone I wouldn't go back to Verizon unless they had an iPhone.
I have had some good CS with both companies and some reallly ****** service with both. In the end I am persistent and always get it resolved, but with both companies it usually takes hours on the phone and speaking to at least 5 different people.
Point blank... all carriers suck and treat their customers like crap.
Just my $0.05...