As an AT&T user, I can tell you that you have to consider how coverage is where you are before you decide which carrier you want to go with. AT&T has far better 3G and HSPA+ speeds than Verizon when it comes to 3G. The thing that complicates things is that Verizon has LTE service in more areas. AT&T LTE is faster than Verizon LTE for the most part, and AT&T has been deploying LTE at a fairly aggressive pace around the country, so as I said, it all depends on where you live. There are a number of areas where Verizon does have better service overall.
Now, on a phone perspective, Verizon is known for butchering GPS, where GPS is unreliable due to the software on the phone and changes made. AT&T is a LOT better, and will let you disable a lot of the pre-installed software, even if it can't be uninstalled, at least you can disable what you don't want.
Now, there are some things about Verizon I personally don't like, and to be honest, I find Verizon to be a dishonest company...the word sleezy comes to mind. A part of this is that I live in a place where Verizon is the local phone company, and have seen them throw $10 services onto the bills of people who never requested the service. It turned out that Verizon threw a "one month free trial" of a service at customers, and if you did not call to say you didn't want it, you got stuck paying for a service you never even knew you had(unless you read your bill). This may not be recent stuff, but with that sort of history, I will NEVER give Verizon a penny of my money, even if they offer LTE where I live while AT&T has capacity issues on my local tower. I'd rather know my bill doesn't change from month to month than have to check my bill to make sure something strange isn't going on.