In the closing credits, the character "Ace", who's always cooking his Louisiana cuisine, is listed as Bobby Hutcherson. In fact, Ace is played by Billy Higgins, who was the drummer at the Blue Note.
Although the movie takes place in 1959, in one scene near the end where Francis and Dale are walking along the East River in New York, one can clearly see the twin World Trade Center towers in the background. These towers were not completed until the '70's.
In 1959 Rico reeds came in a tan box, but the box of Rico reeds that Dale has is the modern orange color.
Although the film takes place in the 50s, one shot of New York near the end shows several mid 1980s Chevrolet Caprice Taxis, along with some cars from the late 1960s and 1970s.
The recording studio is wildly anachronistic. The sort of console in the control room didn't exist until at least the mid 70's. They were much smaller and simpler in 1959, when stereo was in it infancy.
The drum hardware and cymbal stands are 80s vintage, i.e. heavy tubing, whereas the proper period stands would be much lighter and smaller.
Microphone visible at bottom of frame during the nightclub scene between François Cluzet and Martin Scorsese.