Not all 350W units are created equal. We need specs to be able to tell you.
What turkey is trying to say is when Asus says it needs a 450W PSU, they mean for the entire system. Going from memory the 660 is more of a 125W part, while the 660TI is a 150W part. But you need to take the CPU into account as well. My guess is you have a very basic 350W PSU and it won't run the 660 because it lacks the 6pin PCIe plug that's needed. But without specs I'm just guessing.