A controlled-source method has the advantage that the frequencies used are perfectly known, which means that the response due to a given excitation can be better identified in a noisy environment; natural sources are processed over relatively wide frequency bands, which means that it is more difficult to separate noise and the signal of interest. A controlled-source method implies more logistics and it is difficult to have access to sources with low-enough frequencies. A natural-source signal implies less logistics and possibility to look deeper. However, be aware of the AMT dead-band (more or less 1-5 kHz). Also, when using controlled-sources, keep in mind that the source needs to be located far away (e.g., 5-10 skin depths at least) in order to interpret the data as MT data. Source modeling is possible, but it adds significant computational overburden.
Reference: Chave, A. D. and A. G. Jones, 2012, The magnetotelluric method: Theory and practice, Cambridge University Press.