I find that there is a difference and it's not very subtle. Believe it or not, it was the comedy Phoenix Nights that taught me the difference in a single line. When people say "live life to the fullest", it means doing everything you can that you enjoy all the time before Him up there loses his rag with you.
Existing is like how an inanimate object exists, or a sound exists - it doesn't do anything, its purpose is to do a single thing in its lifespan before being forgotten and buried beneath the lonely sands of time. Many people, myself included, neglect ourselves by doing absolutely nothing and thus if we don't contribute anything to society or ourselves, we are simply existing.