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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.

Stochastic versus deterministic model:

• ARMA is a stochastic model in the sense that the dependent variable -- the realizations of the stochastic process -- is specified as a sum of a deterministic function of lagged dependent variable and lagged model error (the conditional mean) and a stochastic error term.
• GARCH is a deterministic model in the sense that the dependent variable -- the conditional variance of the process -- is a purely deterministic function of lagged variables.

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.

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 with focus on the conditional mean.
• GARCH is a model for the realizations of a stochastic process with focus on the conditional variance.

Stochastic versus deterministic model:

• ARMA is a stochastic model in the sense that the dependent variable -- the realizations of the stochastic process -- is specified as a sum of a deterministic function of lagged dependent variable and lagged model error (the conditional mean) and a stochastic error term.
• GARCH is a deterministic model in the sense that the dependent variable -- the conditional variance of the process -- is a purely deterministic function of lagged variables.

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.

Stochastic versus deterministic model:

• ARMA is a stochastic model in the sense that the dependent variable -- the realizations of the stochastic process -- is specified as a sum of a deterministic function of lagged dependent variable and lagged model error (the conditional mean) and a stochastic error term.
• GARCH is a deterministic model in the sense that the dependent variable -- the conditional variance of the process -- is a purely deterministic function of lagged variables.

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 with focus on the conditional mean.
• GARCH is a model for the conditional variance of a process.

Stochastic versus deterministic model:

• ARMA is a stochastic model in the sense that the conditional mean dependent variable -- the realizations of the stochastic process -- is specified as a sum of a deterministic function of lagged dependent variable and lagged model error (the conditional mean) and a stochastic error term.
• GARCH is a deterministic model in the sense that the dependent variable -- the conditional variance of the process -- is a purely deterministic function of lagged variables.

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 with focus on the conditional mean.
• GARCH is a model for the realizations of a stochastic process with focus on the conditional variance.

Stochastic versus deterministic model:

• ARMA is a stochastic model in the sense that the dependent variable -- the realizations of the stochastic process -- is specified as a sum of a deterministic function of lagged dependent variable and lagged model error (the conditional mean) and a stochastic error term.
• GARCH is a deterministic model in the sense that the dependent variable -- the conditional variance of the process -- is a purely deterministic function of lagged variables.