The differences are on the one hand quite simple, but on the other hand quite hard to explain in non-technical language...
The spot healing brush is intended to be used almost surgically - you can vary the spot size, and apply it only where you need it. It reduces the amplitude of the area you've selected but it doesn't do this just by reducing the volume - the spectral shape is also smoothed around the edges - as you can see if you do before and after comparisons.
Auto Heal does pretty much the same thing, but across the entire audio spectrum in the time selection you've made. Sometimes this is appropriate, sometimes it isn't. You have to learn to make the choice, and it's entirely material-dependent; there is no blanket 'optimum' solution. What you are aiming for each time is to eliminate as much of the problem as possible, whilst leaving as much of the original alone as you can manage.