To me, the major difference between the two is that Caliban is actually being physically enslaved. By contrast, Montag's loss of freedom is more mental. That means that Caliban's desire to become free is one that is focused on physical things. He wants the spirits to stop tormenting him and he wants to be free of the physical labor that he is forced to do. Montag just wants to be free in his mind.
The other thing I would say is that Caliban was once free and so has a better idea of what he wants. Montag has never in his life been free and so his desire is less focused and more theoretical.
The interesting thing about the members of Montag's society is that they are enslaved by their own free will. It wasn't even the government's idea to get rid of books and thinking in general. Rather, the government only had to facillitate a process that was begun by society itself. Montag was a part of that society, and he was seemingly content until he realized that there was so much more than what his society had to offer. He gains his mental freedom by stuggling not only with the government and other members of society, but with himself. He is so unfamiliar with the world of knowledge and books so that he must struggle with his own blissful ignorance before gaining his freedom. Caliban, on the other hand, is physically enslaved and longs to be free from his captors, who subdue him by physical, rather than mental means.