Recap: You have swapped out the power cable and video cable. Both work on another system and the cables from the other system make no difference on yours. You plugged your monitor into a different system unit and it worked and you plugged a different monitor into your system unit and it worked. Is this correct?
First thing to try is re-seating the video card in your system unit (unless it is onboard). Then check that none of the pins/sockets on the VGA ports on the back of the monitor and on the system unit are bent/deformed or have anything in them.
Is it a CRT? Were your monitor and the other monitor identical (the pinouts could be different if they are not, even though the video cable is the same)?
What is the manufacturer and model names of the two monitors and the video card in your system unit?