Well,
all the funda of complex nos is basically for mathematical simplicity (or is it?)!
I think the advantage of using exponentials is that instead of getting the frequency components in sines and cosines, we get them as magnitude and phase which are more important from analysis point of view.
The sines and cosines are just brothers out of phase. They are no different... But to understand the signal as a combination of sine and cosine is not that easy. Instead, if you say that this signal is made up of a certain complex exponential (which you have to imagine!) of this MAGNITUDE and this PHASE, it is easier to understand. If you see the Euler's equation, you will understand the subtle conversion from real sines and cosines to complex exponentials...
A lot of imagination is required...
Regards,
asymbian.