Let's first define the methods:
Thumb Under (TU): the thumb is brought under the hand in order to pass the 3rd or 4th finger for playing the scale. The thumb has movement up and down (to play the key), and to the sides (to do cross-over, to transition between group of notes). Here is how it looks like:
Thumb Over (TO): the thumb is treated like the other 4 fingers, the thumb has only up and down movement (no lateral movement), so there is no thumb cross-over. The transition through group of notes in the scale is done with arm and wrist movement. The term is sometimes disliked because the thumb doesn't really go over anything, but that's what we'll call it for simplicity. It can also be seen as avoiding the Thumb Under method. We can see TO in action here.
Which are the particularities of each method? When one should be preferred over the other? Seems that some teachers go so far to prohibit the TU method and use strictly and exclusively the TO method. Why? This implies that it is not about preference, those teachers do believe the TU method is detrimental to their students. So, what's bad about the TU method? What does TO has that TU doesn't? And similarly, what does TU has that TO doesn't?