The law is to curb poaching. Here's more from the NJ hunting regs:
No person shall have in possession a deer, bear, migratory game birds or turkey that they did not kill unless it has a tag bearing the name, address, telephone number, license and permit numbers of the person who killed the deer, bear, migratory game bird or turkey. Those in the business of processing deer, bear or turkey, that is butchers, taxidermists, etc., shall keep a ledger of all of their customers. Each customer's name, current address, day and evening telephone numbers, hunting license number, and possession seal number of any black bear, deer or turkey being processed shall be included in the ledger. The ledger shall be made available for inspection upon request during the hours of 9 a.m. to 5 p.m. by any law enforcement officer or employee of the Division of Fish and Wildlife.
Possession of naturally-shed deer antlers is legal. Possession of naturally shed deer antlers is legal. Parts of deer possessed, other than shed antlers, must be from lawfully harvested deer. Proof of lawful harvest (tag or seal) should be retained for verification. Road killed deer are intended only for consumption; antler possession from these deer is not legal.
So basically, shed antlers are fine. Antlers from shot deer are fine if you have proof of lawful harvest. Road kill antlers are verboten.