I have been doing research for myself online this afternoon and have found the following site that was linked from the USDA's site. The USDA says that you should obtain a health certificate but refers you to this link to the regulations for each state.
California for example states they do not require health certificates but does require rabies vaccination proof.
North Carolina, for example - where Cooper is going, does require a health certificate and will honor that certificate for a period of 30 days (which is a good thing since Cooper will be there until July 13th).
Louisiana also requires a health certificate and proof of rabies vaccination (which is usually noted on the health certificate) within 12 months unless the pup is 3 months or younger. This is where Cooper is leaving from tomorrow . . . our home state. So . . . I guess my vet was speaking for Louisiana whereas it is a regulation for our state.
http://www.aphis.usda.gov/vs/sregs/