There would almost definitely be a Verizon Wireless logo somewhere on the iPhone’s case, probably on both the front and back. There may be separate Verizon music, video, and app store icons that you can’t delete. A built-in feature may be disabled at Verizon’s request because they want to sell you their own version for an additional monthly fee. Verizon may want a cut of any iTunes or App Store revenue from on-device purchases, the cost of which Apple would probably happily pass along to either users or developers. (My guess: Developers.)
This is Verizon we’re talking about. They might “save” us from some of AT&T’s problems, but they’ll bring their own.
It’s easy to think that the grass is always greener away from AT&T, but keep in mind that these are cellular carriers: massive oligopolists that don’t give a shit about us. Their phones are ARPU vending machines, first and foremost, not communication tools. Cellular carriers are only a small step above cable and phone companies in the contempt and disregard they show for their customers.