Applies to any network structure.
$
The difference is quite obvious, a cyclic network allows a loop back to a node higher in the tree. Hence this can allow infinite loops (bad), or recursive processing/memory (good).
An example of a cyclic network would be the road network in the UK, one can leave London by one road/branch and by following a route of nodes end up at London again and the starting node (or any other ndoe previously visited. A stupid driver with no memory could go round and round in circles.
A non-cyclic network could be a biological teee. YOu have the trunk, and then main branches, and then secondary bracnhes, twigs, and then leaves, but no backwards connections from leaves to twigs, twigs to branches, etc.