Different mechanics can create a very different feeling for the player, as the "
System Does Matter" essay discusses.
Of course a huge example for me is the Theurgy system, which is not a reskinning of the magic system. Theurge players have to be concerned with what time of day it is, whether they've eaten, and their local environment: does this ceremony require a fire? Is it raining? What types of activity does the ceremony involve? Will it attract attention?
The system itself helps to keep the player focused on what their character is doing. The fact they have to perform time consuming ceremonies is a big deal, story-wise. Every ceremony is an opportunity for the GM to introduce story concerns.
If theurges were just reskinned wizards, you would have none of this. You could add fluff to make up for maybe a few things but almost all of these concerns would be "lost in translation".
I think psionics could make an impact for the players if it used different mechanics. One example from D&D 3.5 is the idea of "
psionic focus". Certain actions and abilities are only possible if you "have focus". Certain abilities require you to "expend your focus", which means you can't use abilities that require focus until you take another action to regain your focus. If you have 0 psi points, you can't get focus, which could work in conjunction with a staff-mana like mechanic.
Just adding concerns about psionic focus could make psionics feel very different from wizard magic, even if psionics reused wizard spells for most of its effects.