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.