Advantages/disadvantage of zero padding
Advantages of zero padding:


If length of your sequence doesn't correspond to the size that can be handled efficiently with FFT routine (usually powers of prime numbers) then you might want to add some extra zeros to the nearest power in order to get the maximum speed-up. In worst case you double the memory you need for your signal.
Adding zeros is equal to interpolating samples of your spectrum with $\mathrm{sinc}$ function. Therefore you will find it looking more smoothly. It is not affecting frequency resolution in any way. 
If some peaks were split between two bins, these can be interpolated to some extend and you might be able to retrieve some amplitude information thanks to that.
If you are trying to plot frequency response of some FIR filter from it's impulse response then you need to add zeros to it. Otherwise you will get as many samples as the length of filter is. Please see here
If you are trying to convolve your signal with some pattern using FFT, then you need to pad your signals with zeros to the appropriate length. Otherwise result of convolution will be incorrect (replicas in frequency domain will overlap). Please see here


Disadvantages of zero padding:

Can't think of any. Oh maybe except of people believing that it will improve their frequency resolution in some magical way - I knew PhD's saying that...