The battery life on the Droid Bionic is rather bad. It is also several months old, and a bit outdated already.
The Droid Razr has a much bigger screen than the iPhone, but consequently, it is not as pocket-friendly, even though it's very thin. The iPhone runs on Verizon's 3G data network, and the Razr runs on the newer 4G LTE network which is much faster. Check the network data maps on the Verizon Wireless website and see if the areas you frequent are covered by 4G.
If you choose the Razr, be aware that the Razr Maxx is going to be released soon. It's slightly thicker, but has about double the battery life.
I don't think a case is really necessary for the Razr, since the case is made of Kevlar and the screen is Gorilla glass. On the other hand, I wouldn't dream of not putting a case on an iPhone.
I have an iPhone 4, but I've been coveting the Razr. However, I'm in the middle of my 2 year contract, so I guess I'll keep what I have until I can sign a new contract and get a subsidized phone.
I'd go into the store, and play around with the phones. Tell the salesperson that you're just there to look at the phones, and aren't necessarily looking to buy yet.