A trojan is a program (or thing) that promises one thing and delivers another.
Remember Helen of Troy . . . one side rolled up a large wooden horse as a gift, the other side rolled it inside the gates and then the baddies jumped out.
The internet equivalent is something that promises pictures of Anna Kournikova or an eCard from your friend for example, but when you run the program not only do you get what you think you were getting, but you also load up something else in the background.
As Clubman says - the badness delivered by the trojan could easily be spyware - they are not mutually exclusive.
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