The exact difference is arguably a matter of debate, and not every set theory formalizes classes, but intuitively, classes refer to things like the class of elephants (species), the class of white objects, the class of integers, etc. Sets are (naively) permissive enough to mean any collection of objects. It's a philosophical problem and the foundations of mathematics are very philosophical.
Indeed, mathematicians/logicians/philosophers like Leśniewski (Tarski's doctoral adviser) rejected the axiomatic set theories of his day on philosophical grounds as poorly formulated hacks that merely bandaged over the ultimate sources of the paradoxes1 and betrayed the basic intuitive notion of Cantorian set which he understood to be free of paradox; to him, singletons and empty sets were absurd notions2. His published papers use the word "class" (which, IIRC, was interchangeable with set), so in his case, no distinction was made. In order to preserve his notion of Cantorian set, Leśniewski formulated what he called mereology as a replacement for set theory. In place of set membership, he defined the part membership relation ($x \sqsubset y$, read "$x$ is a proper part of $y$") which can be used to define the ingredient relation ($x \sqsubseteq y \iff x \sqsubset y \lor x = y$). If $x$ is the only ingredient of $y$, then trivially $x = y$, or, $x$ is the class $y$ (no singleton). If $\neg \exists x [x \sqsubset y]$, then $y$ is undefined (no empty set).
In Leśniewski's case, his replacement for set theory was grounded in a nominalistic metaphysics that understood classes as mereological sums. The class of mathematicians is a real, concrete object in the world which is the sum of all mathematicians and this class overlaps the class of logicians, another sum, since some mathematicians are also logicians. Some may find this understanding of classes odd, but as far as sets are concerned, it can at least as equally be argued that the singleton and the empty set are absurd notions. The mereology does not, on its own, require the nominalistic interpretation, but it's worth mentioning that it played a role in his work. Also worth mentioning is that mereology was further grounded in two further systems which, together with mereology, formed a cohesive, rigorous system, namely, ontology (where "is" is defined) and protothetic (which entails first-order logic).
Furthermore, mereology is immune to the stock paradoxes that plague(d) set theory. Take Russell's paradox, for instance, understood as the class of all classes that don't contain themselves. Remember, $x$ is nothing but the sum of its parts. If we suppose "contains" means the ingredient relation, then trivially, $\forall x [x \sqsubseteq x]$. Therefore, all classes contain themselves and thus the question of the class of all classes that don't contain themselves is incoherent since no such classes exist. On the other hand, if proper parthood is understood to define "contains", then $\neg\forall [x \sqsubset x]$ by virtue of the antireflexive property of $\sqsubset$ (given as an axiom). Therefore, the class of all classes that don't contain themselves is merely the sum of all individuals (the universe). The case for proper parthood is a little more subtle. For further discussion, I refer you to Leśniewski.
In any case, the whole point I was making is that "it depends". Some systems define class formally, some don't. Some make no distinction. But what is most important to understand isn't this or that particular formalization but the basic question these set theories are trying (or should be trying) to address and that is they are trying to pin down and formalize the intuitive notions of class and set which is related to older metaphysical questions about universals, species, genera, etc. Without understanding that basic problem they're trying to solve, what it's all about, you'll be wasting your time, aimlessly and arbitrarily looking at this or that axiom without any sense of what set theory is, why it is, what makes a good system, etc. It will appear as if there exist these mathematicians that mysteriously come up with axioms for no reason and the rest are supposed to merely check them for consistency or accept them as incomprehensible dogmas. You'll be like an amnesiac walking down the street without any idea of where you are or what address you should be heading to. All too common, I fear.
1 Russell's paradox concerns the set of all normal sets. A normal set is a set that doesn't contain itself. The question is whether the set of all normal sets contains itself. The paradox: if the set of all normal sets doesn't contain itself, then it must contain itself because it's normal, but it would contain itself and thus wouldn't be normal and thus couldn't contain itself, which means it would be normal, but...
2 I recommend reading his translated works published in Topoi. The process of reasoning is interesting in itself.