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”at home”, when used literally, talks about the being at the physical position where you live.

”in the home” - the emphasis here is on the activities, other actions and so on at the place which someone normally lives. The emphasis is not whether this is a flat, apartment, house and so on - but on this place being the residence of the person and the associated thing happening there.

So,

”I’m at home most nights” = ”My physical position most nights is the place where I live”.

It doesn’t matter whether it is a flat or house - the important thing is my location - where I normally live.

and

”Many people celebrate Christmas in the home” = The activity of ”celebrating Christmas” is performed where many people normally live”.

or

”The police arrested a criminal in the home of Jim Smith.” = ”The police performed the activity of arresting a criminal where Jim Smith normally lives.” It’s not important that Jim Smith owns/rents a house or flat, but the arrest happened where Jim Smith lives usually.

So, ”Items in the home” is asking about the items you have at home that you use there / perform activities with.

”where you keep things at home” - the physical location of different things where you live.

That would probably help in understanding both. I found this on the net. They express different things:

”at home” it more general and usual. E.g. ”I’m at home.”; ”I left my bag at home.”.

”in the home” means more of like inside the house. E.g. ”I left my keys in the home” (and you are locked out now; at home however would mean that you went somewhere and forgot them at home); ”Where have you been? We waited in the garden for like hours!” - ”I was in the home all the time.”.

I myself would almost always prefer ”inside the house / flat” to ”in the home”.

At home can be used when answering questions far from home, In the home can be used when answering questions near or surrounding of the home