Verizon has the best overall voice coverage in the USA. If you live in, or travel to, the boondocks on a regular basis, Verizon's your best bet. AT&T's voice service is overloaded in certain cities, and their overall voice coverage across the US is lower.
AT&T's data speeds and data coverage tend to be better than Verizon.
AT&T uses GSM, and if you travel chances are that it will work anywhere on the planet (and most of the rest of the planet has better GSM coverage than the USA). If you have a Verizon iPhone, it won't work in many countries. Actually, parts of Asia are the only places where you'll be assured some service.
That said, both are bad. Both companies charge you 2x - 3x what you'd pay for comparable service outside the USA. Both companies charge you $20/month to enable the HotSpot feature in your phone (which is not a service and costs them nothing). Neither company is particularly good at customer service.
Both companies provide your location data, call records, data history, access to your voicemail, and real-time access to your phone conversations to third parties.