Why do you we say a is congruent to b mod n and not equal? When talking about modulos why is that we use congruency and not equality?
From the accepted answer in the post What is the difference between congruency and equality? It states that:

... two figures are equal if they have the same points.
Congruent figures have the same shape and size (informally) but
possibly different points.

Which confuses me since I think 7 mod 4 results in the same value or the same point as 3.
 A: This is mostly a question of taste; but there's very, very good reason for that taste. Indeed, it's important to disambiguate things thoroughly to make sure we're correct when it concerns technical details.
Generally, the core factor where you can say "7 is equal to 3" without the fear mistaken is being in the right context, namely, being in the space $\mathbb{Z}/4\mathbb{Z}$. In this "four hour clock", $7$ and $3$ are indeed equal, but it's not the $7$ and $3$ that you know from $\mathbb{Z}$, it's one element, generally written with a hat, like so: $\hat{-1} = \hat3 = \hat 7$. The way we write this element refers to $\mathbb{Z}$, since we construct $\mathbb{Z}/4\mathbb{Z}$ from $\mathbb{Z}$; so it helps our intuition. But I could just as well forbid you from writing in a way that refers to $\mathbb{Z}$, and decide to write $\mathbb{Z}/4\mathbb{Z} = \{ 0, \alpha, \beta, \gamma \}$.
In the context of $\mathbb{Z}$ itself, $3$ and $7$ are NEVER equal. They are only EQUIVALENT, relative to the equivalence relation of congruence modulo 4. From any equivalence relation, you can make an algebraic quotient, and in the resulting space, the elements now become equal: $7$ and $3$ are both mapped to a single element $\gamma$, and their results are then equal. But you must never lose track of the space/context in which things are happening, since this is ripe for misuse and mistake.
