I went to both last year. Both are huge, Syracuse is bigger, but Carlisle is run a lot better. Syracuse has a lot of different cars too.
With Carlisle, you have a specific classes with assigned areas, and Syracuse is park where you find a spot. Sunday is a bust for both shows, so Friday and Saturday is choice for either, but I see Syracuse has added Thursday and eliminated Sunday, which is probably a smart move. There's no real class structure at Syracuse either and they have awards that are "Best of" or "Celebrity Choice" rather than classes. Expect to see a lot of hot rods in Syracuse, and they have a rolling cut-off that the car has to be 25 years or older.
Syracuse also has more going on after the show, but the cops have been cracking down on burnouts. Since I'm 15 minutes away, I can't tell you the best place to stay in Syracuse, but like Carlisle, you want to get reservations early. Don't stay in downtown Syracuse. Syracuse does have camping available (reserve that early too) and you can park a trailer on the grounds.
Both have good and bad points and both shows are very different. A lot of Mopar guys do both. When the shows have fallen on the same day in the past, I've gone to Syracuse, but that's more because I can sleep in my own bed.