When you tantalize people, you torment them in a specific way — by showing them something they want but can’t have. You could tantalize people with cavities and nut allergies by eating pecan pie in front of them.
The word tantalize comes from a Greek story about a guy named Tantalus who was so evil that the gods of the underworld came up with a special punishment just for him (they were really good at that). They put him in a pool of water that drained away every time he bent down to drink. There was also a tree above his head that whisked away its juicy apples every time he tried to pick one. So, it only makes sense that to tantalize someone is to torture them with what they cannot have.