As others have said - the lanyard on an axe is so you can hang it up. If you have the lanyard around your wrist and it slips from your hand then the axe is going only one way - back to you! Without the lanyard it flies off safely (well, relatively so).
The lanyard on a knife is different. It can be for whatever you want - to hang it up, to stop it falling, to make it easier to retrieve from the sheath. Whatever.