It means you live until you run out of time. No difference, really - there’s just more emphasis on the fact that our time is finite.

It's a question of emphasis. "Living out" emphasizes the idea of "every last day," to the very end. It is parallel to the use of the word "out" in "running out." Of course, once you have the word "final" you have that idea already.

I would say there isn’t much of a difference. However, the difference could be the level of description. Such as the “how”. ‘Living “OUT”’ has a specific purpose; to live out is to spend the rest of your life in a particular way. This is usually expressed at the ending of someone’s life. I found “live out” in another context: She retired and lived out the rest of her days quietly in the country.