"La mie" is the white & soft inside part of the bread.
"La croûte" or "la croute" is the crunchy crust outside.
In France, people like having a nice crust, it’s what you have when you eat a “baguette” for example. On the other-hand, if you buy a “pain brioché” or “pain de mie” there will be a very thin crust but not crunchy at all. I would say it’s only a brown “mie” not really a crust.