(This question is inspired by the question: When do enemies save against the Command spell?.)

I understand that the DM is thinking that the resisting part of the spell come on the target's turn, because they are "torn" between doing the command or their free will.

Is there any mechanical difference doing the Wisdom save on the start of target's turn, instead when the spell is cast?

What possible problem could arise by allowing the target to delay making the save until their turn?

It's about information

The main difference is that if the save is rolled when the spell is cast, then you know immediately whether the spell has worked or not. You can then use this information to inform your decisions for the remainder of your turn. For example, if you issued the command "halt" and knew that the target failed their save, then you know you don't need to move out of that target's melee range for this round, since they won't be able to attack you. In contrast, if the save is rolled on the target's turn, you don't know what the result is until after your turn is finished, so you need to account for both success and failure as you take the rest of your turn.

There are other minor mechanical implications as well, such as giving other characters a window in which to affect the outcome of the roll (as pointed out in Luke's answer), but the difference in information that you have on your turn is the main distinction, since it will be relevant every time you cast the spell.

Yes. One example would be that other creatures may have turns before the targeted creature makes a save. They may use spells, abilities or some other action to affect the outcome of the Wisdom save.