Julie's shoes change to a different style when she walks out to the balcony after singing "California Here I Come".
The Jazz Singer (1927) had its world premiere at the Warner Theatre in New York, not the Winter Garden as depicted in the film (as "The Jolson Story" was a Columbia picture, the change is understandable).
Mama Yoelson was already dead by the time Al became a star.
When Jolie and his family are at the premiere of The Jazz Singer, he hums along with "There's a Rainbow Round My Shoulder". This song is never sung in The Jazz Singer.
Immediately following Jolson's announcement that he'll star in The Jazz Singer, a view of New York is shown at night with an unmistakable Empire State Building seen in the distance with its unique lighting (although its appearance varies somewhat in the daytime view). The Jazz Singer debuted in 1927, but the Empire State Building only opened in 1931, with excavation only beginning in 1930, 3 years later.
The same audience shot is used at two different theater venues.
Jolson flies back to New York for Julie Benson's (Ruby Keeler) Broadway opening in "Show Girl" which opened July 02, 1929, three years after filming of "The Jazz Singer."
Jolson and Ruby Keeler (Julie Benson) were married a year before she appeared in "Show Girl."
When the story progresses to 1927 and for every year thereafter, clothes and hairstyles are not contemporaneous but are from the film's production year of1946.