Inside the motor unit itself there is a drive gear(usually a worm gear) that is most likely made out of nylon/plastic. Eventually, the teeth and/or groves wear down (or sometimes completely off). While this gear may seem to work just fine when the chain is removed, when it's put under pressure with the chain attached it doesn't have enough contact to pull the weight of the door.
It has been my experience that the cost of replacing this gear (with labor) is easily at least half the cost of a new opener. I went for the new opener.