Sounds like it's possible the PSU can't handle it, especially if this is what happened to your old similarly-powered graphics card.
If unplugging the PSU from the mains is what gets the computer to turn back on, it was the PSU that caused the shutdown. If the motherboard caused the shutdown, it will usually turn on then turn straight back off a second later, once the motherboard realises it's been turned on but doesn't want to.
I've taken a closer look at the sticker on the side of the PSU (thanks for the photo). It says it gives out
21A on each of the 12v rails. It sounds like 660 GTX requires
24A on the 12volt rails. If the graphics card had two PCI-e connectors, you could spread the load over two rails, and those two 21A rails would be enough power (because they'd combine to 42A). But since it only has one connector, the PSU needs to be able to give out 24A (ideally more) on a single 12V rail.
I think this may be your problem.
Found someone with the exact same card, PSU, and problem, too:
https://forums.geforce.com/default/topi ... ng-gaming/That person "fixed" his problem by under-clocking the graphics card in the end, I wouldn't recommend that because you're selling your graphics card short. But if you're short on cash it's a short-term solution.
Power supplies are at least much cheaper than graphics cards. If you go looking for a PSU, try and find one with more 12v rail amp-age. 24 is apparently what you need, but you'd need to make sure the graphics card got an entire rail to itself. (Don't connect anything else like HDDs to the same cable the GFX card is using)
E.g.: Exact same brand of PSU, but has 25A on the 12V rail, $60:
http://www.newegg.com/Product/Product.a ... 6817190028E.g.2: Different brand, has 55A on the 12V rail, $70:
http://www.newegg.com/Product/Product.a ... 6817171084Just examples, you can probably find cheaper. Sounds like this is definitely your problem though.