One or the other might be more common for people living in houses in certain areas, but I've heard both and both sound completely fine to me.
I would be less likely to say 'house' if my home was not actually a house. I currently live in a split-level townhouse, and sometimes refer it to as "my house" and sometimes as "my home". When I live in an apartment, I use "my apartment" or sometimes "my home". So, I would say "my home" is more common simply because 'home' includes more types of dwellings than 'house'.
If 'honey' is aware that I'm at home, I would probably just say "when I leave" without specifying what type of residence I'm leaving.