Banach manifolds have found many uses such as, gauge theory (Donaldson theory, Seiberg-Witten theory, FLoer theory), symplectic topology, to name few I am more familiar with.

One great advantage of Banach manifolds over Frechet manifolds is the implicit function theorem which in the Banach context takes a simpler form, and thus Banach manifolds are easier to recognize.

One disadvantage of Banach manifolds over Frechet manifolds is the fact that natural notions of real analyticity are harder to implement on Banach spaces.

One place to learn about Banach manifolds is Lang's Differential and Riemannian Manifolds

Banach manifolds have found many uses such as, gauge theory (Donaldson theory, Seiberg-Witten theory, Floer theory), symplectic topology (Gromov-Witten theory), to name few I am more familiar with.

One great advantage of Banach manifolds over Frechet manifolds is the implicit function theorem which in the Banach context takes a simpler form, and thus Banach manifolds are easier to recognize.

One disadvantage of Banach manifolds over Frechet manifolds is the fact that natural notions of real analyticity are harder to implement on Banach spaces.

One place to learn about Banach manifolds is Lang's Differential and Riemannian Manifolds