there's a couple things that can fry a monitor consistently, dust and a higher than normal voltage feeding the monitor. and, as a third reason, age. stuff wears out. from the sound of your description i'd guess a too high voltage going to the monitor, but if the recently troubled one is as old as the other that burned out, it could just be age. oh, and come to think of it, there's another possibility, heat. i had a monitor once that was getting so hot it was melting the solder inside. i'd periodically have to crack the case open and re-solder a couple places.
electronics do wear out. my worst problem has been with harddrives, but that's usually more mechanical. the motor or bearings tend to wear out. with monitors, especially crt's, heat, dust and excess voltage can chew them up pretty quickly. with lcd's the dust issue isnt usually as great, but can still occur if you block the vents. line voltages tend to vary a fair amount in the U.S. and spikes are an everyday occurrence. so, surge suppressors are always a good idea. back up power supplys shld be checked periodically, also. they can wear down and cause excess power feeds into the computer. i used to have a computer that was supposed to feed an external harddrive 125 volts. it was actually feeding it 190. a power transister had gone bad and was feeding excess juice to the drive, which almost burnt out.
oddly, i dont suspect your video card. are you running an sli rig with two vid cards or just one with two outputs? if you're running an sli rig and one feed is the one constantly causing trouble, then yes, that one card might be suspect. swap it out or switch cords and see what happens. generally, though, there's not enough juice coming from the vid card side of things to do the kind of damage you've got going on. so, i'd suspect the power coming in and feeding the monitor. it's just one of those things you're going to have to track down and if it's just age, it can be tough.
one other thing i've heard from time to time over the years is, with electronics, you're supposed to 'burn the unit in' when you first get it. this means turning the thing on and leaving it run continuously for X number of hours, usually a two or three days, if i remember correctly. this 'sets' the components to a working condition and is expected to give the unit longer life. however, there is another condition in electronics where once you have components that have been working together for a long while and you replace one of them, that new component will not have the same 'set' as the older ones and something tends to go bad quicker, or so the theory goes.
ok, one last note here after reading your problem once again:
The power button would no longer turn the monitor on and off
that, to me, is a very strong sign of a power surge or continuous stronger than shld be power connection or heat. if the power button is getting fried, that's not good. the unit could be shorting out somewhere or heat could be melting something. the first thing i'd do is just sniff the unit and try to detect any burned smells. and, though you didnt mention it, if you've ever heard and crackling or seen any smoke, you've got serious problems with voltage. check your power supplies and line voltages and if you have a UPS (universal power supply), check the juice coming off that, too.