Why is Hanna so snappish and rude to Michael at times?

This goes back to the book, and some of the backstory that is not as illustrated in the movie adaptation.

In the book, it is much broader in the depiction of the physical closeness being much more than the mental closeness, as they do remain distant. During the later phases of their initial relationship, they begin losing that as Michael starts spending more time with his friends, which I believe figures in to her hasty departure without notice.

The second part is the guilt that she feels over her period as a guard at the Auschwitz satellite. During this time, she had the weak and sickly read to her before they were sent to the gas chambers. This is echoed by how she has Michael read to her.

I believe that both of these combine to keep her at a mental distance, while maintaining the physical side of the relationship. As the mental strain continues, and the physical side starts to deteriorate, she leaves. This also sets the stage for Michael's distance in his later marriage.