Aid:
At home no one's likely to come to your aid for a long time unless you're able to make a phone call. Yelling won't help, gunshots probably won't bring aid unless there are a LOT of gunshots. Fireworks, backfires, etc. are all easy explanations of loud noises and houses make good silencers. Most houses have phones that can be relatively easily disabled from the exterior of the house.
In public there's a good chance someone will see and call for help--or maybe even help you themselves. Gunshots in a public place will almost certainly bring aid.
Retreat:
At home, retreat is probably not an option due to family being present. You can't just run out the back door leaving your family to fend for themselves.
In public, retreat is almost always possible and is almost always a good option.
Hostage:
The outcome of being taken prisoner in your own home with your family can be pretty unpleasant as demonstrated by a recent headline.
In public it's unlikely you'll be taken hostage, and even if you are, it's not likely that you'll be taken prisoner in a private place with your entire family. To some extent this limits the possibilities.
Carry issues:
Carry is simple at home. There's no requirement to conceal, there are virtually no legal restrictions. Carrying a full-sized gun unconcealed in a good holster is not hard to do, it's comfortable and practical.
To me it makes sense to be MORE prepared at home than in public. You've got more to lose at home, less chance of rescue, better chance of something VERY unpleasant happening if you lose, and it's easier to carry.