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You can find authoritative explanations of the differences between “equity” and “equality,” but I think the definitions of these words vary, and there is no objectively correct distinction.
However, we can generalize a bit. To say that something is “equal” does not imply a positive value-judgment. Some people are taller than me. That means that our heights are unequal, but it is not an injustice. Nor does making things more equal always improve justice. Procrustes stretched his prisoners who were too short and lopped the feet off those who were too tall to make all their bodies an equal length. That was not an example of justice. “Equal” has a meaning in mathematics (already attested in Chaucer), and when it’s transported to social and ethical contexts, it retains its mathematical flavor of value-neutrality.
It’s true that the word has long been used as a synonym for fairness. Milton:
… till one shall rise
Of proud ambitious heart; who, not content
With fair equality, fraternal state,
Will arrogate dominion undeserved
Over his brethren, and quite dispossess
Concord and law of nature from the earth …
But Milton has to say “fair equality.” Out of context, without such a modifier, inequality may not imply injustice.
In contrast, the word “equity” has a positive valence, whether in the law (a “court of equity”), in ethics, or in social analysis. If something is equitable, to that extent it is fair. The question becomes: What constitutes fairness? Answers vary depending on people’s philosophical beliefs, social roles, and cultures.
In this well-known picture, several things are equal (the heights of the boards that make up the fence, the altitude of the ground at all points, the sizes of the three boxes, the height of all the heads in the second picture). Some things are unequal, especially each person’s body height.
Despite its label, the first situation is not equal in every respect. But it is inequitable according to three reasonable standards of fairness: everyone should get what he or she needs, everyone should have equal opportunities, and everyone should be a full participant in the activity. These standards can diverge–or they may not even apply in some circumstances–but here they converge to rule the first situation unfair.
The second situation, meanwhile, illustrates equality in some respects. All the heads are the same distance above the fence; the fence is level. But that picture also illustrates equity because it meets several reasonable standards of fairness.
In this case, giving everyone an equal upward boost is inequitable, because their needs are different. But that is only one way in which equality can diverge from equity. If one person really deserves or merits more than another, then giving both people the same amount would be equal yet would violate equity. If Procrustes came along and violently made these three people the same height, that would be equal but not at all equitable. And if we hacked a portion of the fence away to let the short kid see, that would be equitable among the viewers but perhaps unfair to the owner of the fence. In fact, we only celebrate the solution in the second picture if we think it is fair to be able to watch a game for free from over a fence.
The main point is that “equity” always requires an account of fairness: what fairness demands in the circumstances. Equality, on the other hand, always requires measurement. Sometimes when a given measure is equal, that demonstrates equity, but sometimes it doesn’t.
See also: we are for social justice, but what is it?; trends in egalitarianism and sorting out human welfare, equity and mobility.