Ganz grob: Boroughs sind historisch eher städtisch gewachsen, districts eher ländlich.
Aber es gibt keine großen Unterschiede. Am Beispiel von Surrey:
The present structure of local government in Surrey dates back to 1973/74, when numerous urban and rural district councils were replaced with nine borough councils and two district councils, and Surrey as a non-metropolitan county council...
District councils could, in theory, pursue measures to get themselves reclassified as boroughs, this would involve petitioning the Queen for royal charter to make the change under Section 245 of the Local Government Act 1972.....The other difference is that a borough council has a mayor while a district council has a chairman. This distinction is only ceremonial in nature.
Why are some of our councils districts and others boroughs? Get Surrey finds out - Surrey Live