You're nearly correct. The natural relative minor shares all those notes. There are other minors that don't.
The concept is that any piece has a 'home' sound. Generally, in a major piece, it sounds at home - at rest - at the point where the root chord/harmonies get played.
So, considering that fact, if the piece was purposely written in the relative minor instead, 'home' would feel to be, as I said, where the root/harmony is at that point.
A piece in C will sound at home on C. In Am, it will be there on Am.
That's one reason why the harmonic minor came about - sharpening the leading note to push better to the root.
You should also consider modes, which use those same seven notes you have in the major scale, but use another, different one as 'home'. Dorian uses the second scale degree/note, and pieces written in Dorian sound minorish, but come home on that second note. Think Irish folk songs as some examples. Check out Mixolydian mode - another folky way to play - neither major or its relative minor, albeit still using the same old seven notes.