Sometimes it makes things simpler (quite a a lot of things in combinatorics), other times it is a tools for nice tricks (I have no idea how I would solved these equations if it were not for generating functions, see the appendix from a Mafia game paper, https://arxiv.org/abs/1009.1031).

Generating functions, Z-transforms are indispensable in probability theory, Physics, signal processing, and now it seems for a good round of Mafia while camping with friends.