I had both, but I only kept the 350. For me the difference was the price ($129 on sale vs. $229), the sharpness of the screen, and the size. The large screen on the 650 is really nice if you want that sort of thing, but it's a tad less sharp due to having the same resolution as the smaller 350.
I bought the 650 on Sony's USA web site by being persistent. Keep checking back often and you'll eventually get one.
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