The most obvious differences relate to their relative power relations. Prospero asks; Ariel answers. Even when Ariel is trying to remind Prospero of his obligations to free Ariel, there is a hesitancy.
A related difference is that Ariel speaks to Prospero, while Prospero speaks to Ariel, the world, and perhaps a host of other invisible spirits.
If we turn to content, Ariel, while descriptive, focuses on the tasks at hand, and is more precise. Prospero, by contrast, refers to a much wider range of activities, discussing all of his many activities. Ariel focuses on the present; Prospero discusses power, and reflects on the past.