In my experience in analysis, basically the only place where it is actually important to distinguish sets from proper classes arises when one wishes to invoke Zorn's lemma to locate a maximal object in some non-empty partially ordered set $X$ in which all chains are bounded (e.g. to create a maximal proper subspace, a maximal filter, a maximally defined bounded linear functional, etc.). Here it is crucial that $X$ is "small" enough to be an actual set (e.g. it is a collection of subsets of some space $V$ that is already known to be a set, or a collection of functions from $V$ to yet another set). For instance, one cannot use Zorn's lemma to construct a maximal set in the class of all sets, or a maximal group in the class of all groups, or a maximal vector space in the class of all vector spaces, despite the fact that in each of these classes, any chain has an upper bound (the direct limit). (Such maximal objects, if they existed, would soon lead to contradictions of the flavour of Russell's paradox or the Burali-Forti paradox; not coincidentally, one of the standard proofs of Zorn's lemma proceeds by contradiction, using the axiom of choice to embed all the ordinals into $X$, which can then be used to set up the Burali-Forti paradox.)
To put it another way: regardless of one's choice of foundations, it is clearly mathematically desirable to be able to easily locate maximal objects of various types; but it is obviously also desirable for the existence of such maximal objects to not lead (or mislead) one into paradoxes of Russell or Burali-Forti type. ZFC, with Zorn's lemma on one hand and the set/class distinction on the other, manages to achieve both of these objectives simultaneously. Presumably, many other choices of foundations (particularly those which are essentially equivalent to ZFC in a logical sense) can also achieve both objectives at once, but I usually don't see these points emphasised when such alternative foundations are presented in the literature.