Appraiser estimate is higher than body shop estimate?

I'd ask the body shop for a written estimate, so you can compare line by line what the difference is, before doing ANYTHING. If it's all apples to apples, let the appraiser write you a check, take it to the body shop, get all the work done, and keep the difference. As it's the other guy's insurance, you CAN request the check be made payable directly to you. Keep in mind, the appraiser might be adding in rental car coverage to his estimate - that's why a line by line comparison is important.

You can take the car to the body shop and if additional damage is found you or the shop just calls the insurance company back for a supplemental estimate. This is done all the time because a first estimate usually does not pick up additional damage that is found later.

Everything Ginger and mbrcatz wrote is 100% correct. Most likely the body shop either wasn't too thorough or low-balled the estimate because they thought you were price shopping. What is likely to happen is that you will bring the car in, the body shop will find all this extra damage when they take the bumper off, and you'll get a call asking for more money or for the name of your insurer so they can call them and advise of the additional damages. In some areas of the country independent appraisers are not paid for their time to do follow-up estimates for additional damages (called "supplements") so they write the estimate on the high side. $2,000 is a pretty big difference though. The body shop likely doesn't have something on the estimate that they actually will need. It's not illegal to keep the difference in such a situation but it can cause it to get a lot more complicated. It's possible this will work out for you but it's equally possible with that large of a difference that the body shop will end up discovering everything the independent appraiser saw as well as stuff he didn't see. In this instance you should go back to the shop and give them the appraisal from the independent appraiser and ask them to explain the difference by showing you the damages the appraiser saw that the shop did not.