They are the same, and yet they are not (pretty Zen, huh?)
You've discovered "relative" chords. There are actually a bunch of chords that when played could have different names. The musical context will tell you which is right. In a half diminished chord, to take an extreme example, any of the notes could be the root.
As to why we need the different names, imagine you were playing a I-IV-V, but the chords were called C (I), A(IV), and D(V). The Letter name of the chord indicates which note is the root, while the part of the name after the letter tells you the harmonies and how it will "feel" in a progression. It also lets you know if the chord is as you would expect in the key you're playing, or if it's been altered, and a whole host of other stuff.