My mother has a driver's license but doesn't drive.

Literal

Mother [topic-は] driver's-license [object-を] has-[ongoing-が], driving [topic-は] does-not

持っているが、運転はしない — が joins two contrasting clauses: 'has (it) but doesn't drive.' 運転はしない — は on 運転 adds contrastive emphasis: 'as for driving, she doesn't.'