Overlap testing has a lower processing cost, is easier to implement yourself and is what you would use for collision between objects if realism is not important to your simulation. One thing to note is that 'bullet physics' or the collision of very small and very fast objects with thin surfaces won't always behave as expected. This can be resolved by decreasing the time-step of you physics simulation, but at a higher cost.
Intersection testing is generally more computationally expensive and harder to implement, but excellent optimized implementations like Bullet Physics make it easy for beginner and expert alike to use this type of testing. It is much more realistic and bullet physics are supported by virtue of the way collisions are calculated. So if your game mechanics revolve around physics, using a library like bullet can give you the realism you need.
All in all, you should use overlap testing for simple games that do not need the additional accuracy (and overhead) of an accurate physics simulation and intersection testing for everything else.