$ACC^0$ is a natural complexity class.
1) Barrington showed that computation over non-solvable monoids capture $NC^1$ while over solvable monoids capture $ACC^0$.
2) Recently, Hansen and Koucky proved a beautiful result that poly-sized constant width planar branching programs are exactly $ACC^0$. Without the planarity condition, we of course get Barrington's result characterizing $NC^1$.
So the difference between $ACC^0$ and $NC^1$ is group-theoretic on one hand and topological on the other.
Added: Dana, a simple example of a solvable group is $S_4$, the symmetric group over elements. Without getting into details, any solvable group has a series whose quotients happen to be cyclic. This cyclic structure gets reflected as mod gates while building a circuit to solve word problems over the group.
On planarity, one would like to believe that planarity may impose restrictions/bottlenecks in the flow of information. This is not always true: for example, variations of planar 3SAT are known to be NP-complete. However, in smaller classes, these restrictions are more "likely" to hold.
In similar vein, Wigderson showed NL/poly=UL/poly using the isolation lemma. We do not know how to derandomize the isolation lemma over arbitrary DAGs to get NL=UL, but we know how to do so for planar DAGs.