Equality has come to mean "equality of opportunity" unless it is specifically used with some other modifier. So, for instance, if everyone with a SAT of over 1500 is offered a scholarship at Stanford, that would be "equality of opportunity": there is a clear goal, and if you pass it you get the reward, regardless of who you are.

The idea behind wanting equality (of opportunity) is that we structure parts of our society to reward merit in order to have a society that produces meritorious individuals, to the benefit of everyone. If we don't have this, we're squandering people's potential, which is both unfair to the individual and wasteful to society.

Equity has come to mean "equal representation across identifiable groups, especially racial and gender groups". So if you set the SAT threshold at 1500, and you get only 1% of black students reaching the threshold, you get worried about the equity of setting the system up this way.

The idea behind wanting equity is that we have good reasons to believe that individuals of different groups have very similar distribution of qualities, so if a system was actually fair, you would expect to see (absent other factors) that the proportion of groups would reflect the proportion in the broader population. The thought is that when you don't see proportional representation, something went wrong somewhere and it would be good to fix it.