Watson falters an instant as he opens the lid of the musical box after removing it from the biscuit jar, causing the lid to mostly fall closed again for a second before he lifts if open fully. Yet the tinkly music continues to play seamlessly, without any interruption from having the lid closed again momentarily.
Dr. Watson refers to having met the King of Bohemia. This refers not to the land of the Czechs, but to an imaginary place of the same name, created by Sir 'Arthur Conan Doyle' for the Holmes adventures.
When Sherlock goes to see the auctioneer to ascertain who purchased the music boxes, he reads the purchasers address from the log and says its on "Hampton Way" In the next scene he visits the house on Hampton Road. He also mentions the "correct" street, Hampton Road in a subsequent scene.
The goof item below may give away important plot points.
When Holmes is shown the first music box, he is allowed to hear just its first few bars. Still, he then whistles the entire tune as if he knows it. However it soon develops that the tune is unique, and Holmes could never have heard it before. He could not have whistled it entirely.