The previous answers by @Buck Thorn and @M. Farooq are very good, but not quite complete. The main advantages of FT UV-Vis spectroscopy would be 1) accuracy of wavelength (or wavenumber) determination and 2) better spectral resolution of bands, relative to conventional UV Vis spectroscopy. But it is very hard to do well, and very expensive.
The disadvantage is that source shot noise, which is white noise, and source 'flicker' noise, which is low frequency noise with power spectral density proportional to $1/f^\alpha$, get modulated by the moving mirror. The modulation frequency for any given wavenumber of light is directly proportional to the light's wavenumber value. But the source noises also get modulated.
The absorbing specimen absorbs the modulated light, as per the absorber's absorption bands, resulting in an interferogram like the one shown in M. Farooq's answer. Then the FT does the demodulation, i.e., 'decoding', resulting in the spectrum of the absorbing specimen. But the modulated source noises also get decoded and cause a reduction in the signal-to-noise ratio (S/N or SNR). The noises on the large spectal bands are the problem: the white noise spreads throughout the spectrum, thereby lowering the SNR and making it harder to resolve the small bands. The source flicker noise also causes trouble by making 'band shoulders', as it were. Thus, the hoped for advantage of high wavelength accuracy (and improved spectral resolution) gets very significantly compromised by the decreased SNR.
For more details, see E.Voigtman and J.D. Winefordner, “The Multiplex Disadvantage and Excess Low Frequency Noise”, Applied Spectroscopy, 41, 1182-1184 (1987). Also see the comments posted after Buck Thorn's answer.