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
$${\mathbb{Z}^d}$$
. 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
$${\mathbb{Z}^d}$$
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
$${\mathbb{Z}^d}$$
with d ≥ 4.