check drivers and how new is the moniter
Look in "Device Manager", is there a yellow question mark showing against the monitor or perhaps your "Hibernate or sleep mode" settings are causing it and it won't wake. Is the BIOS set for "plug and play". Just some suggestions.
My bad, obviously you can't check it if it won't display
I'm a bit confused on whether you are using the same VGA cable, when it works in the other computer. Could it be the cable or maybe the VGA socket in the monitor is faulty.
Try turning down the hertz and resolution on the other computer with VGA
But as long as your using a straight VGA cable without any type of converters(IE DVI to VGA) you should be fine)