I have never stayed at Vero, but have been to Hilton Head.
With a 5 month old, I would not worry about entertainment for the baby. It's more about things you can do with him/her that are easily adaptable. If you do want to go to the beach, then I would say Vero Beach, since getting to the beach in Hilton Head is more difficult. If your baby starts to get cranky on Vero Beach, it is a lot easier to go back to the room than it is to do so from Hilton Head.
But Hilton Head does have a lot of things you can do, that if necessary, can easily be modified based on your baby's mood or needs.
If you plan on going out and about off of the resort on Hilton Head, then definitely a rental car with a car seat is needed.