When the family and Vinnie go to the baseball game, they clearly sit behind a row of seats (with fans in them). The following shots of them singing the National Anthem has them sitting behind a railing, not behind a row of seats with spectators.
After Vinnie marries Shaldeen in Reno, he is talking on the phone with Barney - and then puts Shaldeen on the phone to speak with Barney. When they're done, Barney asks Shaldeen to put "Vinnie" back on the phone. Shaldeen knows him as "Todd" not "Vinnie."
When Barney tells Vinnie "Keep your nose clean, capisce?" and mispronounces "capisce" as "capisse," Vinnie corrects him, pronouncing it "capeesh." The correct pronunciation is "capeeshay." "Capeesh" is how the word is pronounced in an Italian (slang-like) dialect. Although Vinnie could be using the alternate pronunciation, this would be inconsistent with the scene where he meets his mother and speaks fully articulate Italian with her.
On the plane, when Barney and Vinnie are flying to New York, Vinnie is paying the stewardess for his alcoholic drinks. When he raises his left hand, to pay the stewardess, he is handcuffed to Barney's right hand. A few seconds later he is telling Barney about his tipping "philosophy". He asks Barney for a pen and Barney raises his right hand, which is no longer handcuffed to Vinnie's left hand. Then, a minute later they get off the plane and are handcuffed together again.
(Possibly deliberate errors by film makers): The FBI would not send Vinnie to live in a town/city where there are other mobsters in the Witness Protection Program. Vinnie's reunion and subsequent crime spree with his old pals would be the very reason why they would avoid placing in a city with other mobsters.
When Vinnie asks Barney when his wife left him, Barney replies, "October." That couldn't be correct, because she left with ballplayer Wally Bunting when he got demoted to the minor leagues in the middle of the season. The major league baseball season typically ends on the final weekend in September.
When Barney spends the night with Hanna, she explains that the kids are not home, thus keeping their budding romance secret from her kids. The next morning Hanna's ex-husband barges in to the house expecting the kids to be ready to go. Both Barney & Hanna appear to just be getting up in the morning, and could not have kept up the ruse had the kids been present or just returning from her mother's. The ex-husband is expecting the kids to be ready to go, and their absence would be a serious breach of a joint custody arrangement - especially for Assistant District Attorney Hanna.