First - You're somewhat married to your power source. Most cases, this means keeping the fridge in the car. But when camping with smaller kids, it can be difficult for them to retrieve a drink and you need to keep the car unlocked. I have cargo drawers in the back of my LandCruiser, and it can be a pain to get stuff out of the fridge due to the height. I have started bringing a second battery so I can leave the fridge out as I would a cooler.
Second - You need adequate power. Your car's battery will run most fridges for a day, maybe two. But how far do you want to push it before you harm your battery or are left stranded? Or do you factor in the cost of a dual battery setup? Having a second battery, not even a dual battery system, has alleviated some of this anxiety, however.
Third - this one is really minor, but it does bug me a bit. I love grabbing a drink out of the ice cold slush in a cooler. I don't know why, but something is missing when I grab a drink out of the fridge. It's too sterile. I want the beer label to peel off from soaking in ice water for two days.:sombrero: