Well, if by contact glue you mean something like model airplane cement then no, they are different. "Contact cement" in puppet building usually refers to an all-purpose contact cement like LePages or Barge (which is what the Muppet builders use).
Small Nerf balls are usually sold at Toys R Us and WalMart or you can search online to find a vendor. Some people also use hard syrofoam balls or plastic spheres (clear plastic Xmas balls) that are commonly sold in craft stores.
That said, I feel it's necessary to go off on a bit of a rant re: contact cement...
I would stongly recommend against using contact cement. Despite what you may have heard, hot glue can work fine for most projects and contact cement is not really necessary unless you are doing a professional job (and many Pros just use hot glue).
Contact cement is toxic and the fumes are extremely dangerous. You cannot use it indoors without a proper fumehood. In addition to being totally environmentally unfriendly, there are very serious sideffects to inhaling contact cement fumes including severe headaches, nausia, sore throats and all sorts of other nasty stuff. Exposure to the fumes without proper protection will also compromise your immune system for several days and make it easier to get ill. You can also cause damage to the inside of your nose.
If you do have to use it, it's best used outside or in the garage if you can keep the gargage door open. You have to wear a proper respirator with the appropriate filter (usually an organic vapour filter). Eye protection and latex gloves should also be worn, because as bad as it is to inhale the stuff, absorbing it through your skin is even worse. The cement also stains anything you get it on and it's generally a pain in the a@@ to work with.
If you are serious about working with materials like this you should first take a WHMIS course to learn about how to safely work with chemicals and what to do in an emergency.