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When we talk of class models and set models, is there a need to talk about them separately? What would be an example?

What I can think of is that a class cannot technically be talked inside a model or a universe that a class is in without some tricks, and thus this applies for class models too without a trick. I think I have heard of it somewhere, but my memory fadedd away.

Bools
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One main difference with class models is that it is much more difficult to construct and then work with the set of true formulas of a class model, compared to a set model. The set of true formulas (i.e. the complete diagram, in model theory terminology) is used very often when we want to work model theoretically.

For example, consider the consistency proof of Peano arithmetic (PA) within ZFC: we consider the model $\omega$, and prove by induction on formulas that $\omega$ is a model of PA, and therefore PA is consistent. This induction relies on a construction of the set of true formulas of the model.

We know that ZFC cannot prove itself consistent, by the incompleteness theorem, but why can't we take a class model of ZFC and show that it satisfies ZFC, in the same way we take a set model of PA and prove it satisfies PA? The reason is that we cannot work with a truth set of a class model in the same way as a set model.

If we move to a stronger set theory like Morse-Kelley set theory (MK), we can now form the truth set of formulas of ZFC for a class model. This is how MK is able to prove that ZFC is consistent, by showing in MK that $V$ is a class model of ZFC. But we cannot form and work with the truth set for formulas of MK on a class model, within MK itself.

So, although we may realize that the existence of a class model of a theory proves the consistency of the theory, this is not always formalizable in the set theory we are working with.

Carl Mummert
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Henkin's Theorem (the completeness theorem): A theory is consistent if and only if it has a model. Here we mean a set model.In ZFC you cannot quantify over proper classes, so you cannot assert "There exists a class such that ... ".

DanielWainfleet
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    In ZF or ZFC a definable class can be considered to be just an abbreviation. For example,Godel's constructible class L has a long definition. "X belongs to L" means that X satisfies a long list of conditions. But in ZF or ZFC it is impossible to say "There exists a definition...." A language that allows unlimited use of sentences about sentences produces paradoxes. – DanielWainfleet Aug 08 '15 at 21:57