There's really no difference. Mainly I use a machine at home instead of a data center VM because that's just one less bill to pay. I have two GPUs in there which would be very expensive on public cloud.

I guess one reason people might gravitate to home hosting is owning your own disks, the tinfoil hat perspective. You can encrypt volumes on public cloud as well, but it's still on someone else's machine. They could take a snapshot of the heap memory and know everything you are doing.