It all boils down to the root note. You can play the c major scale, but if you use E as your root note, you have E phrygian. Your asking whats the difference? It's the root note, the root note is the primary note in whatever you play. You'll usually start there, repetitively go back to it, or end there. Each of the modes give a different color to the music, this color is called phrasing.

So if your friend is playing a D minor chord and he says play D dorian. If you think of C major your phrasing will be off. Although it's the same notes, you find C, as a root, doesn't mesh good over a D chord.

This is why if you look in Jazz Improv books, they'll give you chords. maybe all in one key, but under each chord they're ask to play the mode that goes with that chord. this is because of phrasing. And good jazz soloing depends on it.

Try this if you can.

Record yourself playing a rhythm with a D minor chord. Play it back and play scale runs in C major. Next try it with D dorian? Which sounds better?