If we apply the raising (creation) operator to $Ψ_n(x)$ and the apply to it the lowering (annihilation) operator, we get $Ψ_n(x)$ times a constant. Does it physically say something? Can we get any intuition out of it?
Now, i know that the wavefunction times a constant does not mean anything physically, but we we did the same thing at a wavefunction that is in a superposition of two number eigenstates, then the constant factor in front of each eigenstate will change, so we will get a different wavefunction since it will now "consist of different amounts of each eigenstate" than before. So, i think that the non-commutativity has a physical significance.
Note: This question refers to he cases in which $Ψ_n(x)$ is a number eigenstate.