Yes, there is a difference...
In a nutshell:
TOP QUALITY:
Super Aramith Tournament & Super Aramith Pro-Cup (61.5mm)
Between these two sets, there is no real difference in quality, weight, material etc. These are the balls used in top level competition. The Pro-Cup have the spots. Any "Super Aramith" set is going to play well and last a good long time.
MED QUALITY:
Aramith Premier (60.3mm)
This is a fine "Classic American" set meant for people with older tables. They are slightly smaller in diameter as they are the old standard 2 3/8in diameter.
The Premier formula balls have a tendency to not wear as well as the Super Aramith formula. But both are excellent sets made from phenolic resin.
"WHY BOTHER?" QUALITY:
Sterling
I have no clue what they are made of, but for 20 bucks I am pretty sure I do not want to find out. They could be made out of an old Bakelite formula - they could be made out of Acrylic. I would almost bet money that they are too light and do not meet tournament specifications.
Natural Elephant Ivory
The only reasonable thing to do with a set of Ivory balls is to display them in a case. Like wood, it can warp with changes in temperature and humidity. Unlike wood, you really can't make anything out of it these days unless you are recycling small pieces or it comes from somewhere other than an elephant's mouth. Playing with warped ivory balls is a bit of a waste of time and if you crack one while you are playing, their collector's value will be reduced considerably.
good luck!