If I were invincible, but not immortal, what would be the mechanism for my eventual death? No one ever really dies from old age. People have heart attacks, they die elderly of cancer or infection. Would my heart just suddenly stop on my 80th birthday?
Assuming my memory would be able to handle the increasing burden placed upon it, I think I would choose immortality. I would use make-up to age my face gradually, and when that became too much of a burden, I would fake my death and start over somewhere else. If I didn't get caught in the first few hundred years I would probably get good enough at it to escape detection from then on.
I would have to think about living healthy, not to live longer, but to be more comfortable while I live. I would be immune to disease, but does that mean I could just trash my body with the sort of abuse that only has long-term consequences, like over-eating and alcohol abuse, and not get sick? Or would I be sick, but just not die from it?
I think those of you who mentioned the evolution of humans had a good point. However, if I were immortal, I could have hundreds, thousands of children, and eventually, the human race would all be related to me. At varying distances. I would be a force delaying human evolution, until we were so genetically incompatible that I could no longer breed with them.