Because it could quell a lot of our fears.
For instance, I have a fear of losing one of my limbs. Particularly, one (or both) of my arms. If I were immortal - or just invulnerable - then that fear would be gone because I could never lose either.
Note that this is the Baccano style immortality I'm talking about. The one where you can a) live forever and b) when something is cut or removed, it just all comes back to you. Arm chopped off? A few seconds later, all the blood comes back into you and the limb reattaches itself. That would be awesome.
Honestly, I don't want true immortality on this Earth because unless we go somewhere else, the sun is eventually going to explode. And then there's the whole bit about the Heat Death of the Universe/end of the universe in some manner. It would, however, be nice to have the aforementioned Baccano immortality at about 27 years old for a few thousand years, or a million or so, just to see where humanity is going.
And, of course, so I could play all the newest video games at my leisure. Also not needing food would save me a whole lot of money, and not needing sleep would save me a whole lot of my already-considerable time, enabling me to earn more money!
On a final note, everyone should go watch Baccano now, no matter your opinion on anime (Hint: The performances are excellent and the humor isn't bad at all). Genius, that show.