Pros
I was an intern at AEI with one of their scholars as a graduate student. I had good opportunities to do interesting research, review papers, and help brainstorm talking point for public appearances. Interns are treated really well by staff and scholars, in my opinion. They get opportunities to tour different orgs. and departments around the DC area. The gourmet 3 course meals are pretty amazing. You'll gain weight at AEI if you're not careful.
Cons
There's a strong conservative ideology at AEI and very little else from along the rest of the spectrum. I've found that AEI's reputation on the outside isn't as good as people at AEI think it is. Many people see it less of a think-tank and more of an ideology-center. So, if you want to pursue research and you're already firmly rooted in a conservative position, then this might be a good stepping stone for you. If you're a moderate or liberal person, AEI's a good place to be exposed to to understand what it's like and get the viewpoint, but you'll probably not want to stay here long.