According to this article
Quine has criticized higher-order logic (with standard semantics) as "set theory in sheep's clothing". Quine's criticism focuses on the lack of an effective, sound, complete proof theory; he argues that this makes HOL not a "logic". Shapiro has responded to this criticism, arguing that the additional semantic expressiveness can offset the lack of a proof theory, and arguing that a "logic" need only have a deductive system or a semantical system, but perhaps may not have both.
a. Why does Quine say "set theory in sheep's clothing"?
I tend to agree with Shapiro on the tension between deductiveness & expressiveness.
b. But are there useful results which naturally use the higher-order expressiveness either in their proof or statement, apart from the categoricity of 2-logic as opposed to 1-logic.
Since higher logic with Henkin semantics reduces to typed 1-logic it seems essential to keep full semantics.