Edit: I realized the answer was lacking and have thus provided a more precise answer (see below -- or maybe above). I have edited this one for factual mistakes and am leaving it for the record.
Different focus parameters:
- ARMA is a model for the realizations of a stochastic process imposing a specific structure of the conditional mean of the process.
- GARCH is a model for the realizations of a stochastic process imposing a specific structure of the conditional variance of the process.