Difference of time domain waveform obtained at each sampling frequency

It's different.

When the sampling interval changes, the digital frequency corresponding to the analog frequency will also change accordingly, so its measurement will also change, and the sampling frequency is directly related to whether the original signal can be recovered.

FFT technology is not a patent of real-time spectrometer, and it is also used in traditional frequency sweeper, but the FFT technology used in real-time spectrometer is quite different from it.