Fourier transform(FT) is the generalization of Fourier series. In FT you get frequency spectrum of a function f (if FT exists for f) and instead of spectrum you get discrete frequencies if f is periodic.
Note: Both FT and fourier series are approximations of f
I'd like to clarify that FT and FS are not approximations. Both are mathematical transforms that allow perfect reconstruction of the original function. For example, what is the Fourier Series of cos(x)? Hmmm, seems exact to me.
Where the approximations come in is when you have an infinite series or a transform that extends to infinity. Then, you may decide to truncate the series or band-limit the transform, when using them to reconstruct the original function. Even if they are not infinite, you could still truncate/band-limit. Hence, you could say that a truncated FS or a band limited FT are approximations.