What's the difference between these two words? When do we use the first and when the second?

During is a preposition and while is a conjunction.
During modifies things and while modifies actions.

Few examples:

During our stay (thing)
While we were out (action)

During the coffee break (thing)
While we had our coffee (action)

I think you'd be better at teaching English than me

I wouldn't know how to explain that.

But you are indeed correct.

Haha, cheers man! ;d