Loosely,
Hairy ball theorem is about the derivative of a mapping at a single point in time. (Imagine slowly deforming/mixing the input configuration t to obtain the output configuration.
Brouwer's theorem can be thought of as being about the integral of a continuous family of hairy balls over a time interval.
It's not exactly the same because the assumptions about differentiability are different.
It's not exactly the same because the assumptions about differentiability are different.