Longing and crushing are somehow related, as they both wear off after a while - at least if not somehow reinforced.

Longing, however, can apply to about anything you desire,where crushing is typically focused on a member of your preferred sex, and (some say) it is at it's heart about the thin to non-existant chance of sex with such person. So some say.

But I'll tell you, I am really longing for a shiny female robot to do my bidding, bring me cups of espresso late at night, and I would imagine that I could also have a crush on such a robot. mmm. some kind of Barbarella mixed with Emma Peel robot.