I've decided to put this into an answer because it would be unclear in a comment.
I can second Mark's example, however I would point out that the cocycle z lies in degree 8 and not 6. There's another one, w in degree 11 whose purpose is to kill y^2, which is 0 in the abelianisation. Not to mention xz in degree 10. The resultant commutative algebra seems to be of the form k[x]⊗k[V], where V has some kind of algebraic structure, I'm guessing a (co)lie-module after a suspension, but I can't work out what it should be.
I also have an interpretation of all of this. In working out an associative quasi-free presentation (TW, δ) we're actually working out the bar homology W of the algebra (actually including the A-infinity coalgebra structure as well). But there's a decomposition of the bar homology of a commutative algebra known as the λ-decomposition (see for instance Loday's Cyclic Homology book). If we were to take the "middle" piece W' of this decomposition we would get the commutative bar homology (and it's associated L-infinity coalgebra structure). And this is just what we need to get the resolution of the original algebra as a quasi-free commutative algebra (SW', δ'). All the extra bits (the V of the example) come from other pieces of the λ-decomposition. I think that this should impose strong structural conditions on (SW, δ'').
Going back the example we can work out that z is in the second part of the decomposition, although I'm not sure about w.
So what this means for the various homotopy functors between the homotopy categories in question I'm not sure, the following is speculative: It seems to me that it indicates that they're not full. There are homotopy morphisms from k[x]/x^2 to another commutative algebra that are not homotopy commutative algebra morphisms. They should fit into the λ-decomposition and their position there should indicate just how "not commutative" they really are.
Final note on "not commutativity": this notion can be made more rigorous using operads, but in a hand-wavey way it just means to what level the higher homotopies of the commutativity condition hold.