There's one big advantage to a Verizon contract iPhone - they're SIM unlocked for international use, while AT&T will only do it after your contract is up. This means you can use local SIM's while travelling if you so choose with your new Verizon iPhone 5S.
I have similar Verizon vs. AT&T coverage issues in my area (the former is generally better), and this unlocked SIM advantage was one of the reasons that swayed me over to Verizon when I bought my iPhone 5 last year.