How do you distinguish between an identity and an equation?

[Mark44]

If ≡ was replaced with =, wouldn't it also be true that the coefficients of their respective terms are equal?

No, because the two equations would have different meanings. Let's look at the equation in my example a couple of posts ago.

As an identity:
2 $\equiv$ (A + B)x + A

As an identity, the equation above has to be true for all values of x. In my previous work I solved for the constants A and B, and found them to be A = 2, B = -2.

As an ordinary equation:
2 = (A + B)x + A
If we interpret the above as an ordinary equation (not an identity), given values of A and B, we could solve for the value of x that makes the equation a true statement.

The work would look like this.
2 - A = (A + B)x
$$\Rightarrow x = \frac{2 - A}{A + B}$$
If someone tells us the values of the constants A and B, we can determine the value of x.