I think that's the biggest difference for most. If you can amortize the hardware costs, then 'burst usage' is cheaper at home to a degree, because you are paying a fixed monthly rate elsewise. Overall thought for most, it is likely cheaper to use the cloud than at home, but really depends on what you want.
> because you are paying a fixed monthly rate elsewise
No, you would pay usage based rates with API, in this case. I have exactly one fixed monthly rate for the 6 AI models I have tokens available for.