Yeah they are the same notes, C major and D Dorian. Although in practical application, they are different and they do have a different tone. It's no more apparent than playing a song in the key of C major, and then playing one in the key of A (natural) minor. Those two scales have the same notes but when you use either one they sound quite different. It's all in how you use them, chord progressions and/or melody. By knowing the difference it's opens up alot to your music and your technique as a player.

Think of it as, the root note is a color and the scale you use is the shade of that color. There is only one C note as there is only one Red color, but there are many shades of red, as they're are many different scales (or modes). You are right they are the same notes, but I would suggest looking into modes. Modes have alot of great uses, and to get them you have to also be able to look at them as being different.