The movie takes place in 1962; the young sons are clearly elementary school age, at least. The older son, Nick, was born in 1959 and Frank in 1962 so, in reality, were actually three years old and an infant when the story takes place.
At the beginning of the movie, the font used on the Copacabana entrance canopy (a variant of Futura) and the font on the poster announcing that the club would be closed for two months (Arial) did not exist in 1962, and would not exist for at least twenty years thereafter.
In the year this film takes place, Seventh Ave was one way heading south. The cars are parked facing north. Additionally, background buildings across the street are incorrect; the buildings that were there in that year are still in place.
Tony criticizes Don for not being familiar with several black musicians whose music they hear on the radio while traveling, one of the artists being Aretha Franklin. By the fall of 1962 Aretha had had only six chart hits, none of which reached higher than #37, so it's doubtful that an Italian from the Bronx would have been very familiar with her music.
Tony talks to the stage manager in Indiana and complains about the piano not being a Steinway. Tony mentions Dr. Shirley's name and tells him he is the performer. The stage manager says "who is that?" and then makes a racial slur a few seconds later, but no one told him Dr. Shirley was black.
In the scene with a wrong piano - not a Steinway - provided for the concert, the piano lid is opened the wrong way. The correct way would be to flip the front part first, and then to raise the lid on the stick. But maybe it was done deliberately to illustrate the stupidity of the stage hand.