You already know that first-order logic has a completeness theorem. That means that we can determine validity in first-order logic by looking at deductions - it makes proof theory possible. In second-order logic with full semantics, because there is no completeness theorem, to study things like validity we end up having to answer questions about the power set of the domain.
Here's an example. There is a sentence $\phi_1$ in the second-order language of ordered fields that characterizes the real numbers, up to isomorphism, in second-order logic with full semantics. There is another sentence $\phi_2$, in the language with just equality, which states that the domain has cardinality $\aleph_1$ (that is, any model of $\phi_2$ in second-order logic with full semantics has a domain of that cardinality). Now in order to show that $\phi_1 \to \phi_2$ in this logic, we would have to prove the continuum hypothesis, and to disprove that implication we would have to disprove the continuum hypothesis (this is because $\phi_1$ has only one model up to isomorphism).
Examples like this give us a sense that studying second-order logic with full semantics comes down, in many cases, to studying set theory. But if that's that case, many people say, why not just study set theory, as with ZFC? Studying set theory in the guise of "logic" only seems to obfuscate what's going on.
Moreover, for those who want to use the logic for foundational purposes, it is unattractive to pick a logic that seems to already have the answers to set-theoretic questions like the continuum hypothesis built into it - this goes against the idea that "logic" itself should make a minimal number of ontological assumptions.
This sort of argument was made in detail by Quine, who called second-order logic with full semantics "set theory in sheep's clothing". Not everyone agrees with this, and many people do use second-order logic with Henkin semantics as a way to keep the expressiveness without including the set theory. But the dominant opinion accepts Quine's argument.
I also recommend "The Road to Modern Logic-An Interpretation" by José Ferreirós, Bulletin of Symbolic Logic (2001), 441-484. This paper has a very nice historical study of the development of what is now called first-order logic.