Spyware is things that companys place on your system as part of a download or as a cookie. Its harm is that it does "call home" sharing with them all the info they have aquired about your buying habits etc... but it also tends to cause system problems.
Now a trojan is a device similar to a Trojan Horse in Troy... its a means in which hackers can travel inside your computer.
I hope this answered your question...
Your need is to keep your system free of virus , worms , trojans & spyware
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