Verizon has a decent sized LTE network. It's getting to the point where if you're in any well populated area, you'll have 4G service. They are still expanding, and have a lot of work to do before their 4G coverage has the same reach as their 2G/3G network, but they're the only carrier that is even close to having a full next generation network. The price is high, but you get great service in return. Their software updates may be slow, but they also tend to have fewer issues with software updates than other carriers. T-Mobile and Sprint are the only two fast carriers when it comes to updates, and both AT&T and VZW are on the slow end for product updates and also they tend to have more preinstalled junk than other carriers as well. Customer service is hit or miss. It really depends on what you're having a problem with. Their software/hardware troubleshooting leaves a lot to be desired, but it is still better than the rest on average from what I've heard. They tend to be much better with account issues from my experiences, but I don't have any recent comparisons to another carrier for that to say if it's better or worse.
AT&T will have better battery life on their phones as they are full GSM, not a CDMA/GSM (LTE) hybrid network. The nice thing is you can unlock a phone and use it on their network, something you can't do with VZW, and this opens up a lot of possibilities for phones. They tend to be a little less controlling when it comes to devices they sell as well (branding, locking, etc.). Cost is on par with VZW, and you shouldn't have a problem with coverage as long as you aren't travelling through an unpopulated area. However, coverage isn't the same as what VZW offers, and the network can be over saturated in some areas. They have more "4G" than VZW, but most of this is HSPA+, note LTE. Speeds are comparable, but LTE has the potential for much faster speeds in the future, where HSPA+ is pretty much being maxed out.
Overall, I wouldn't say that one is significantly better than the other. Neither is a good option if you are looking to cut costs, and customer service is probably about the same for most things. A lot of customer service issues are going to really depend on who you talk to, as there are always times you get the person that doesn't know anything and doesn't care, they're just there for the paycheck, and other times you get someone that cares or knows what they're doing, and it solves your problem fast. One thing good about VZW is that all of their call centers are in the US, not sure about AT&T. I think the choice should really be, do you want to pay for the network and go through the hassle of switching carriers just for a phone.
