The thing with authorization is... well... you can have authorization to go to a public park. But carving up a park bench is very much not authorized. You can be "able" to do something when the public has access to it. However the ethics of altering property that belongs to somebody else tends to lean from anywhere from "okay if you have permission" to "heavily illegal".

Anybody can install spyware on a public computer, anybody can install spyware when given permission by the property owner. Installing spyware on a public computer (that you do not own) without permission of the owner is very much illegal.