First of all, to me, a house is just a structure while a home has emotional connotations. These can be good or bad, since everyone had different home experiences.

I usually ask people to "Come over to my house" not "Come to my home". Or if I'm describing something about it, I use "My house is really small" not "My home is really small". But when I get in the door, I shout "I'm home!" because it's more than just "I am now in this building", I mean something more like "I am in this building and I can relax now and pet the cat and take my shoes off and have a snack...Come and hug me!" Similarly, "leaving the house" is just going out for errands or something, but "leaving home" is permanent, it implies that you're moving out on your own.