Is Asteraceae and Compositae the same family?
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.