I wouldn't base my decision on things like restaurants or shopping when we're talking about a choice between Bellagio and Caesars. The two places are right next to each other. It's not like we're talking about the Stratosphere and Mandalay here. Besides, a ten minute walk outside back to your hotel after dinner would be refreshing.

Decide based on the things that you can't change with a short walk - the room you get, price you pay, and other non-tradable amenities like the pool.

To me, the properties are somewhat equivalent (Bellagio's "5th star" is way overrated). I'd make the decision based on the specific price and room type. That is to say, if I'm looking to spend $X per night, I'd look to see which one would get me a better room. Caesars lowest level rooms aren't as nice as Bellagio's, but if you're comparing, say, an Augustus Tower room at Caesars versus a standard room at Bellagio, go with Caesars.

Also look at what deals you can get from a Virtuoso or FHR.