For instance, using the Laplace transform, we are able to take a time signal and convert it into its frequency domain counterpart to avoid differential equations, but what do we get with Convolution?

I keep on hearing that Convolution in the time domain is like multiplication in the frequency domain. That's great, but what value does using Convolution have?

I have seen a lot of animations of a square pulse travelling over two signals overlapped, but I don't know its significance.

Thanks,

JP