Jess sings to an audience in California in the middle of the movie. At the end of the movie he is singing to an audience in New York (it is assumed from the progression of the film) and it is quite obviously the same audience and venue. A number of audience members are present in both scenes. There is a woman with glasses wearing a vest and white shirt, a man in the middle of the audience with a checked cap, and a large man having a great time down front clapping very excitedly.
When Jess first sits down to write Love on the Rocks in his bathrobe there is a menorah on the table near a beer can. The next shot of the table a minute later reveals the menorah is gone and the beer can is turned.
In New York at the end of the movie when Leo asks Jess to sing in his father's place, Molly and Jess argue at the piano. Both Jess and Molly are wearing wedding rings. Later that night when Jess goes to reconcile with his father he is no longer wearing a wedding ring.
When Molly gets in Eddie's car and has him listen to Love on the Rocks on her tape, she has Eddie turn the radio on and the volume up. The radio clearly has some kind of talk or news station playing and when Molly puts the cassette in, it locks down into the player like just about any tape deck would. However, when Eddie takes the tape out later, his arm motion clearly shows him pulling the tape out like an 8-track (no gesture of him pushing the eject button then taking it out) and the radio station doesn't come back on either.