We use ‘live’ to talk about permanent situations or long periods of time.
- Where do you live? (= What is your current address/country of residence)
- When I was younger I lived in Scotland, but now I am living in Germany.
- Most Australian people live in cities.
We use the word ‘stay’ for temporary situations.
- How long will you stay at your sister’s house?
- Have you ever stayed at that hotel on the hill?
- Where did you stay while you were at the conference?
- When I first moved to Scotland I stayed with friends until I found a place to live.