For people who have had both: How does AT&T realistically compare to Verizon?

Just to put it out there: I'm sick of how locked down Verizon's network is compared to AT&T's and Tmobiles. I've wanted a new phone for a while now, but of course Verizon hasn't gotten a decent phone in forever. And they are notorious for taking longer than other carriers for pushing updates (or just releasing phones in general. Looking at you HTC One).

Moving to AT&T would allow me access to a large LTE network and also access to upcoming Nexus devices (as well as any Google Play edition phones). But I'm worried about coverage. I travel for work, not all the time, but somewhat frequently so I need to have a good cell connection. It's one of the reasons that I'm still on Verizon (plus unlimited data on a family plan). Though, I am almost certain that my parents don't want to buy unsubsidized phones to keep unlimited data so if they were to upgrade their phones, I would also lose unlimited data anyways.

Usually I would just deal with the phone selection because AT&T's network wasn't as good, but they have been expanding their LTE network pretty rapidly and they seem to have "regular 4g" in most places, which I'm sure is much faster than Verizon's 3g. So back to my original question: How do they compare? Any advice or anything is appreciated, thanks.