How come FFT window give half size frequency bands when filter can seems to have an infinite precision? I don't really understand maths behind filtering but use it lot in music production and i'm a total newbie to DSP.

Usually parametric filters let you filter frequencies with a precision that seems almost infinite. But when using a FFT or DFT the taken window size gives n / 2 + 1 frequency bands. 

Are the two related ? if so how can a filter get so much precision when Fourier Transform doesn't?

That may seem a dumb question but when I look at spectrograph in sound softwares they always looks much more precise than the buffer I've choosed (usually 256 samples).
The DFT is a tool to represent the spectrum of any signal. Usually for an N point DFT only N/2 output coefficients are required because for a real signal the DFT is symmetric about the centre.

No filter has infinite precision, neither infinite bandwidth nor numeric representation. A filter with enough precision can be chosen for an application to seem smooth enough.

The DFT though doesn't have anything to do with a filter. It simply represents the spectrum of the signal at it's input.