Using function symbols in first order logic forces us to define "terms" inductively, which makes many proofs longer and much more tedious. Of course, function symbols simplify matters when trying to use first order logic to describe things, but on the surface it seems to me they could be replaced completely by relations: Instead of $f$ use a relation $R_f$ such that instead of writing $\varphi(f(x))$ write ($\forall x\exists! y(R(x,y))\wedge R(x,y)\wedge\varphi(y)$. Now use $f(x)$ as a shorthand notation, so you can use it in "real life" but avoid it in proofs.
I guess I'm missing some deep neccecity here, but what?
(The same goes for constant symbols, but they don't really complicate things as function symbols do).