Ok guys, let me help you on this confusion.
Carom is a short-version of
carambole a/k/a
carambole billiards : the game you play with 3 balls on a 9ft or 10ft pocketless table. you make one point when the CB hit both OBs in one stroke.
Anywhere in the world except the US:
Billiards is the general terms for all games you play on a table with balls and sticks.
Billiards = Carambole Billiards + Pocket Billiards + Snooker +.....
Pool only mean "the swiming pool" or something similar to that, absolutely nothing to do with our game.
So, That's where all these confusions come from.
In the US:
Pool = pocket billiards
Billiards = carambole ( the game I mentioned above )
Carom sometimes uses also to refer Carambole Billiards.
For all that,
billiards is the way americans call carambole (carom).
Playing carom, the objective is using the CB to hit ,at first, one OB and then the 2nd OB. The principle is the same with the shot we play sometimes in pool when you play the CB off one ball to make another ball.
That's why we use the term "carom" or "billiards" to refer that kind of shot.
What's the different between them ? NOTHING, Americans use "billiards", others (countries) use "carom". And then Americans heard the word and started using it. And then some of them think that the word has different meaning.