When you take the integral of displacement, you calculate the total area displaced over time, and when you take the integral of velocity, you calculated the displacement.

It's all physics 101 really!

However, what we see on a sample by sample basis is that summation (integration) and differencing (differentiation) both look like 90 degree phase rotations of the original signal; but in opposite directions. Just as the derivative of sine is cosine, and a sine-wave shifted by 90 degrees is also a cosine; calculus on periodic signals changes the phase of the signal.

Furthermore, the signal envelope, which is a positively valued function, might be a better description of the amplitude intensity versus time. Here is an illustration of signal envelopes on seismic traces.

If you integrate velocity you get displacement. So, integrating tells you the displacement of the ground relative to its original position.

If you differentiate velocity you get acceleration. So, differentiating tells you how much the ground is accelerating at any given moment.