General rules here for me:
Hackberry, Celtis occidentalis, will have lighter, duller leaves to the American elm's, Ulmus Americana, dark and lustrous.
Hackberry bark is a light gray with corky warts, A. elm is dark brown with intersecting furrows.
Hackberry's have fleshy rounded drupes and A. elm's have round, winged samaras.
Hackberry's almost always have hackberry nipple gall and I have yet to see A. elm's with anything like that (however, that might vary in your area).
Structure: could be THE decisive id...Hackberry's are free spirits with a rangy, asymetrical (read rather unkempt) appearance, whereas the American elm will be vase shaped and majestic.
Don't know if any of that helps...
Sylvia