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Why can't you move diagonally and would it make a difference if you do allow it?

I think this is kind of just a design choice. By explicitly stating CAN or CANNOT move diagonally, you lock it in and can design other mechanics (such as how far you can move in a day) around that choice. And it's not an unreasonable choice. If you've ever been out in the wild and tried to move directly diagonally from one area to another on a map without entering the areas bordering on the side, it just doesn't really work that well. If you assume there are roads, terrain, etc., they would all make moving purely diagonally pretty unlikely.

If you change it, I don't think anything will break. You will have PCs that can move a lot farther with less chance of an encounter or other scene. As a GM, if I looked at my map and there was a clear road crossing the corner of a sector into the diagonal one, I would probably allow this (i.e., make it work with the fiction/maintain the verisimilitude). But I also might decide to have something like a walker swarm or a faction with a roadblock right on that corner.