Specifically, nation states set their own foreign policy and have their own military. Usually APTs are associated with military or intelligence agencies, which only nation states have, though some (e.g. the Bears in Russia) aren't formally tied to the government (as much as being allowed to make money on the side.)

I favor APTs as a term, since it denotes any actor with lots of technical means and a political/strategic rather than financial agenda. Many APTs belong to nation states, but that's not always true!

The confusion seems to come from most people seeing nation and country as synonyms. The older traditional definition of nation has mostly faded except in the context of the modern use of the word nationalism, but even that has been perverted to mostly mean extremists.