This is a very domain specific question and the answer depends on your workflow and the application you are designing your algorithm for. Certainly, there is a difference in FFT and CWT before and after merging. In fact, there are specific algorithms in the literature for a correct way of merging frames of FFT and CWT data.
You cannot compare when you would need either of cwt or fft functions. If you scroll down to "Outputs", you will see that both of these functions give different outputs and the answer to "when to apply cwt/fft" is an abstract question.
This link discusses one of the approach to combine consecutive FFTs to create a larger FFT data.
Similarly, this documentation page talks about discrete wavelet analysis of signals with wavelet packet approach.
