Yes. There is a big difference in how these expressions are used. 'At the drop of a hat' means suddenly, perhaps without warning. For example: 'My boss is so inconsiderate. He thinks that I leave on business trips at the drop of a hat.' 'In a heartbeat' is used as to indicate that you would do something without any hesitation 'If I had the chance to go back to college, I'd do it in a heartbeat.' As you can see, we use these expressions to express very different ideas.

They both mean "very quickly". The main difference would be how you use each one. You could say: "I would go to Italy in a heartbeat." AND. "I would go to Italy at the drop of a hat." And you could say: "He gets angry at the drop of a hat." But you really wouldn't say: "He gets angry in a heartbeat." - You could, but it would sound strange. So each one is a little different depending on the situation and what you're referring to.