Something life-threatening is seriously dangerous, and it might even result in death. If someone has a life-threatening illness, it's severe enough that they may need to be hospitalized.
You can reassure your mom about the injuries you got when you were hiking up a mountain by saying, "Don't worry, my sprained ankle isn't life-threatening." In other words, it's not going to kill you. Things that are more likely to be life-threatening include cancer, bad head injuries, and run-ins with grizzly bears — they can indeed threaten your life.