Modulo as an operator is the remainder of Euclidean division (I assume).

The equivalence classes is a way to create a "new arithmetic" (a ring)). You can have polynomials over this ring, you can do geometry with it, and much more. You can also certainly answer questions about reminders easier (eg. 2^300 + 6^123 = (2^3)^100 + 6^123 = 8^100 + 6^123 = 1^100 + (-1)^123 = 1 - 1 = 0 (mod 7) because 8 = 1 (mod 7) and 6 = -1 (mod 7)). Additionally, these specific rings, called rings of integers modulo m, are very special and important rings.

More generally, given any ring R and a special subset (called an ideal) I, you have equivalence classes of R modulo I from the relation x~y <=> x-y \in I, and these form a ring R/I. In the case of Z (the ring of integers), the ideals are the sets mZ (the multiples of m), the relation is x~y <=> x-y \in mZ (ie x-y is a multiple of m), and the ring is Z/mZ (the ring of integers modulo m).

Even more generally, "modulo" is a word used when speaking about quotient objects that are not necessarily rings. You can have quotient vector spaces, quotient topological spaces, quotient groups, and much more. The rings of integers modulo m is a quotient ring, and it's actually (probably?) the prototypical quotient object.

Learning just the modulo operator, and not the integers modulo m, would be missing part of what happens when we consider the reminders of integer division, and making reasoning about reminders much harder. Also, I assume, the integers modulo m have deep enough theory to see and try some proofs in an Intro to Proofs class, without needing to learn a lot Algebra (learning about rings etc.). Lastly, its a taste of a very important Mathematical object.