the difference is in stochastic part or lack of it.

Notice the most recent time index of the stochastic part in your formulation of ARMA: $$X_t=\varepsilon_t+\dots$$ Compare it to GARCH: $$\sigma^2_t=r^2_{t-1}+\dots$$

You can immediately see that in ARMA at future time $t+1$ the disturbance $\varepsilon_{t+1}$ is not yet observed, while in GARCH $r_t$ is already in the past, i.e. observed. Hence, ARMA is stochastic when it comes to forecasting $\hat X_{t+1}|I_t$ and GARCH is not. At time $t$ you already have all information to calculate forecast for $t+1$ in GARCH

the difference is in stochastic part or lack of it.

Notice the most recent time index of the stochastic part in your formulation of ARMA: $$X_t=\varepsilon_t+\dots$$ Compare it to GARCH: $$\sigma^2_t=r^2_{t-1}+\dots$$

You can immediately see that in ARMA at future time $t$ the disturbance $\varepsilon_{t}$ is not yet observed, while in GARCH $r_{t-1}$ is already in the past, i.e. observed. Hence, ARMA is stochastic when it comes to forecasting $\hat X_{t}|I_{t-1}$ and GARCH is not. At time $t-1$ you already have all information to calculate forecast for $\hat\sigma^2_t|I_{t-1}$ in GARCH