In a cyclic group, you can change the dimension of a chart following a specifi order when you left-click on the cycle button.
Field A - > Field B -> Field C -> Field A -> Etc
Or you can right click the icon and select any of the available dimensions.
On the other hand, in a drill down group, you follow a specific path when you select only one value of a given dimension.
If you have the dimension A -> B ->C in a drill down group, you can oly move to the dimension B after you have only one value in dimension A and move to dimension C after selecting only one value in dimension B
I'd say that the cyclic groups are move flexible but drill downs call be really helpful sometimes.