I agree that the discrete wavelet transform is analogous to the discrete Fourier transform, but they are calculated differently, and present very different information in a very different way. I think most people find DWT results much more confusing than FFT results.

If you're very familiar with both of them, it can be easy to underestimate how confusing each is to a newcomer.

They are very similar to a short term Fourier transform (STFT). Both of which permit one to analyze the spectrum of signal over time