If you've ever seen the skull of a horned animal versus an antlered animal the difference between antler and horn is pretty clear. The center of a horned animals horn is bone and forms a single continuous piece connected to the skull, surrounded by flesh and hard keratin (like finger nails). Antlers grow more like a knuckle, not connected to the skull as one continuous piece.

Another horned oddity is the rhino, whose "horn" is more like a specialized fingernail (keratin again) than a true horn. A rhino "horn" is like a compressed lock of hair filed to a point, a hair shiv if you will.

Yet another weird horn like thing is the giraffe. They have bone knobby lumps on their head that are like something between a horn and an antler, being bone fused to the skull surrounded by flesh but without keratin.

Then there are narwhals. I don't know anything about narwhals. But they are cool and have "horns".