Can you explain modes to me?

I was told that the modes are just the major scale with the notes of that major scale with notes rearranged like C major(Ionian) start C end on C.

Then if I wanted to play Dorian of C major I would start on D and on D.And so on…..That is all fine and dandy if that is it, seems pretty simple.

But am I the only one who sees that you still play the SAME notes as the C major scale so why does it matter what mode if the notes are the same??

I was told the only real way to notice modes or “moods” as he put it was to play certain chords of the key in order really hear Dorian or Ionian etc…

So can anyone explain this more to me which is the right way to think of this?

I personally believe in the latter to fully hear and utilize modes you need to have instruments enhancing certain chords/notes to better hear the mode.