We consider the quenched and the averaged (or annealed) large deviation rate functions I _q and I _a for space-time and (the usual) space-only RWRE on . By Jensen’s inequality, I _a  ≤ I _q . In the space-time case, when d ≥ 3 + 1, I _q and I _a are known to be equal on an open set containing the typical velocity ξ _o . When d = 1 + 1, we prove that I _q and I _a are equal only at ξ _o . Similarly, when d = 2 + 1, we show that I _a  < I _q on a punctured neighborhood of ξ _o . In the space-only case, we provide a class of non-nestling walks on with d = 2 or 3, and prove that I _q and I _a are not identically equal on any open set containing ξ _o whenever the walk is in that class. This is very different from the known results for non-nestling walks on with d ≥ 4.