They are either treated or even declared to be synonyms in all the texts using both of the terms that I have ever read.
Just one reference: the nearest _Flora_ on my desk, G. Marconi, F. Corbetta, _Flora della Pianura Padana e dell'Appennino Settentrionale_ , uses the notation _Compositae_ = _Asteraceae_.
It is worth to note that _Compositae_ , 'composite ones' in Latin, refers to the their typical capitulum inflorescence, while _Asteraceae_ comes from _Aster_, a genus belonging to the family, meaning 'star' in Latin, which represents an allusion to the shape of the inflorescence of _Aster_ , and of many other species, which is supposed to look like a star whose rays are represented by the ligules of the ligulate tiny flowers of the capitulum. Nevertheless, there also are species of the family having no ligulate flowers, for ex. _Tanacetum vulgare_.