Not much. The two terms are used interchangeably within the industry.
Strictly speaking, aerospace includes the possibility of the vehicle leaving the earth’s atmosphere, whereas aeronautical does not.
So an aeronautical vehicle operates with earth’s atmosphere only, and aerospace vehicle might leave and return to earth’s atmosphere, and a spacecraft (e.g. the International Space Station) will only ever operate in space. And the engineering that goes into those vehicles will be identified by the same terms.
That is the essential difference between aerospace and aeronautical engineering. And in practice, the two are often used interchangeably.