If you think very, very hard about it, the name "compost" might give you a hint as to why IT IS handled separately...
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Rules for the "compost waste" doesn't apply for making the compost that you use as a fertilizer. Maybe that's why we call it "wet". It's basically anything that is biodegradable. But not everything that is biodegradable is ok for fertilizing...
"compost waste" will usually be used as fertilizer. It depends on the region, but they'll often extract bio gas first and the residue will be further processed by machines and bacteria into compost or fertilizer.
Other garbage will usually be burned and then buried. Maybe partially recycled. But completely different processes.
Residual waste is mostly plastics. Isn't that obvious that you don't want that together with compostable material that will be used to make compost or at least something like biogas?
And it also works in reverse: compostable material is usually relatively wet... so if your residual waste ends up in a waste incineration plant, all that water would be very problematic for burning the waste.
Probably the second is the reason