Does the Fast Wavelet Transform(FWT) produce the same coefficients as the Discrete Wavelet Transform(DWT) if configured for the same depths? Or is the the FWT just an approximation of the DWT?
If the discrete wavelet transform can be implemented with a FIR filter bank, with appropriate extensions, yes, up to numerical precision, coefficients will be the same.
If the discrete wavelet transform possesses a non finite support, then a FIR filter bank implementation would require filter truncation, and the results may differ. On those case, like for spline wavelets, the processing is often performed in the Fourier domain.
There exist IIR discrete wavelets, which may bridge this gap, yet I am not familiar enough to tell.