Why do we use "congruent to" instead of equal to?

• I am a human being.
• You are a human being.

Therefore, I am you: right ? Well, as it turns out, the answer is no. It simply means that we belong to the same class. Likewise, $3\neq7$, but $3\equiv7\bmod4$.

I prefer to use the congruence exclusively (I view the remainder operation as a source of headache even though I realize that it is a necessary evil in computing).

But I use both equal ($=$) and congruent to ($\equiv$) signs together when processing a lengthy calculation in modular arithmetic. My freshman algebra students quickly catch on with my calculations like $$ 12^{3004}\equiv5^{3004}=5^{3000}\cdot 5^4=(5^6)^{500}\cdot 5^4\equiv 1^{500}\cdot5^4=25^2\equiv4^2=16\equiv2\pmod7. $$ IOW, I use $=$ when there is an equality of integers between the steps, and $\equiv$ when I mean a congruence. The power of laws obeyed by congruences is apparent. Checking/following the progress of the calculation is easier this way. Of course, using $\equiv$ all the way is correct also. The $\equiv$ is there as a reminder that in this step we do something that only results in a congruence.

As the students become acquainted with the language of residue class rings, I gradually stop making the distinction between $=$ and $\equiv$ as well as, clarity of context permitting, the distinction between $n$ and $\overline{n}$.

Thinking about what that would look like when done by somebody who is only familiar with binary mod makes me shudder.

It depends on how you define your notation. The standard definition of congruence is $a \equiv b \pmod c$ if and only if $c \mid (b-a)$. The standalone expression $b \pmod c$ is undefined, hence not equal to anything.

On the other hand, some people like to write "$b \bmod c$" to stand for the least nonnegative number $d$ that satisfies $d \equiv b \pmod c$. If you adopt this notation then it is true that $a \equiv b \pmod c$ if and only if $(a \bmod c) = (b \bmod c)$.

The difference between the two is as follows:

Note that (in its usual usage) $a \mod c$ is a function that gives a particular answer, and has nothing to do with equivalence relations per se. For example, we might note that $$ \renewcommand{\mod}{\operatorname{mod}} 7 \mod 2 = 1 $$ That is, $7$ has a remainder of $1$ upon division by $2$. What we cannot say is that $$ 7 \mod 2 = 3 $$ because $1$ and $3$ are distinct values. We can say, however, that $$ 7 \equiv 3 \pmod 2 $$ Since both $7$ and $3$ have the same remainder upon division by $2$.

Why is it important? Well, we often make "substitutions" of equivalent values, and writing "mod" every time you make a substitution becomes unwieldy. For example, we can compute: $$ 5*7*9 - 2*3 + 5 + 3 \equiv 1*1*1 - 0*1 + 1 + 1 \equiv 3 \equiv 1 \pmod 2 $$ If we were to write this with mod as an operator, we'd have $$ [5*7*9 - 2*3 + 5 + 3] \mod 2 =\\ [(5 \mod 2)(7 \mod 2)(9 \mod 2) - (2 \mod 2)(3 \mod 2) + (5 \mod 2) + 3 \mod 2] \mod 2 = \\ [(1)(1)(1) - (0)(1) + (1) + 1 ]\mod 2 =\\ 3 \mod 2 = 1 $$

Equal to means that they are the (exact) same thing. Congruent to means that they have some important property in common, in this case the same remainder when divided by the modulus. The relation, however, must be reflexive, symmetric, and transitive.