When Anne and Otto Frank begin to hum and dance, the closed captioning says "The Blue Danube", which is wrong. The song they hum to is "Tales from The Vienna Woods". These two songs sound similar because they were both written by Johann Strauss.
On the first night of Hanukkah, Anne says it is December 7, 1942 - a Monday - and two candles are lit and blown out. When Otto Frank goes to inspect the break-in, he says it is Saturday and no one will return until Monday. The same two extinguished candles are seen throughout the scene.
When the thief comes the first time, Margot Frank has a cough. She coughs into a handkerchief. In the next shot of her, her father Otto Frank tosses her a handkerchief because she is coughing into her hand.