What is the difference between a jock strap and a cup?

No, Bras do but jockstraps do not. By "cup" I assume you mean "pouch." All jockstraps are sized by waist alone.

A groin cup is something different than a jockstrap. A groin cup it is a hard protective cup that fits around a guy's sexual organs during sports. A jock strap (athletic supporter) is like tight underwear but is designed as only a waistband with an elastic pouch to hold the testicles close to the body so that they do not jangle or get painfully caught out. As for groin cups, there are only 2 "sizes": one for youth, and one for adults. They do not make adult groin cups in different sizes - the size differences on adult groin cup packages are for waist size only. A cup should completely enclose over the top of the genitals as the cushioned rim rests beside the genitals against the groin (above, under, and in the creases between the legs and scrotum). That way any impact will be transfered from the cup to the pubic bone or perineum, thus protecting the incredibly sensitive "balls" (which are terribly vulnerable, but so important to making sperm and testosterone).

It's not just the groin cup that comes in youth and adult sizes. Jockstraps come in youth sizes as well; the pouch is cut smaller to support boys that are less developed.