Difference between a set and a class

Set theory is a mathematical theory, like any other mathematical theory it has a "universe", and the axioms and properties are required to hold within the universe.

In set theory, the objects in the universe are called sets.

But since we don't live inside that universe, but rather work from the outside (in one way or another) we are free to talk about collections of elements from that universe. We can talk about "all the rational numbers which are negative or their square is strictly smaller than $2$", and we can talk about the collection of all sets which have a certain property.

What is confusing in the case of set theory is that sets come to model, mathematically, the notion of a collection of mathematical objects. So if classes are also collections are mathematical objects, why aren't classes sets?

As it turns out, not every collection which we can define form a set. That's the essence of Russell's paradox.

So we limit ourselves to some collections of mathematical objects, and require that they satisfy certain axioms. Classes are collections which need not be in our universe, and therefore don't have to satisfy the axioms of set theory.

Let me reiterate this point. Sets are elements of the model of set theory, and they have to satisfy the axioms, e.g. the axiom of power set. Classes are collections of elements from a model of set theory, but they don't have to correspond to any element in the model, and they don't have to obey the axioms. Just like a real number can be seen as a set of rational numbers, but it doesn't mean that it can be written as a quotient of two integers.