I wouldn't use a spare PC, real waste of energy to run a PC 24/7 just to monitor a temperature sensor.
I can think of two approaches.
First one - buy a ready-made device. Google "differential thermostat" and you'll find products designed to close a relay if the difference in two sensed temperatures exceeds a user set level. These seem to be used in solar water heater systems.
Second one - build that device. Here is a DIY page
http://www.solaffect.com/