Fourier Transform only provides information about the frequency components in a signal. In order to detect faults both frequency and time information is needed -'Which frequency component was detected in the signal and at what time?'
There is no obligation to use only wavelet transform. Start with short-time fourier transform (STFT) and then try other transforms. What matters is joint time-frequency analysis!
I strongly recommend this article: https://web.iitd.ac.in/~sumeet/WaveletTutorial.pdf
Previous comments are valid viz locating sudden changes in current time recordings to locate the fault on the time axis .Additionally we have the freedom to choose basis function other than sinusoids. For example a certain types of fault may produce a characteristic oscillation for a short duration in the fault - time recording that is not a regular sinusoid. In such a case a properly chosen wavelet as basis would help in better discrimination of the fault.