what is the difference between area and magnitude of a pulse?
You have it backwards.

Some mathematician somewhere said 'let's define something over an infinite time, that is only defined by its area. Let most of it be zero, let the peak amplitude tend to infinity, let the duration of the non-zero part tend to zero, such that the area under the curve is constant. And we'll call that a Dirac Delta, aka an impulse'. And a lot of other people agreed, because it's useful mathematically, and it's a good asymptotic model for a number of physical things. 

An engineer somewhere said 'let's define something over a finite time, with a finite amplitude and duration, and we'll call that a pulse'. And a lot of people agreed, because it's useful.

Whether those are good names for the things so defined, or sufficiently different from each other, is a matter for history. That's where we are.

The amplitude, or magnitude, is the instantaneous value.

The area under the curve is the integral of the value over time.

Sometimes we only need to know the area. For instance when a cricket ball hits a bat, the momentum transferred is equal to the integral of the force with time. With leather on willow, the force will be much lower, and last much longer, than with a steel bat and a steel ball. However, the change in momentum for the two will be broadly similar. What matters is not the amplitude or duration of the contact, but the area under the force curve.

Similarly in electrical engineering, anything that depends on an integral, like the current in an inductor after a voltage pulse, or the charge on a capacitor after a current pulse, depends on the area under the curve.