I have both ('08 Am. Std. and a Classic '50s). Tone-wise, I love them both. If I'm playing the Am. Std., I'll think, "Dang, this baby sounds good!" Then I'll pick up the Classic and think, "Dang, this baby sounds good!" A guy at Fender's Customer Service told me that the Classic '50s come with American pickups, even though the bass is MIM.
Comfort-wise, the Am. Std. wins out (for me personally). The width at the nut is 1.65" on the Am. Std.; and 1.75" on the Classic. I also like the back of the neck better on the Am. Std. But that's not to say that the Classic is UNcomfortable. It's just that the Am. Std. is VERY comfortable.
Aesthetically, I like them both. My Am. Std. is Sunburst/rosewood/tort, which I always considered a classic look for a Fender. The Classic '50s is Honey Blonde/maple/gold anodized, which IMO looks awesome. I'd have to call it a draw.
I know it sounds like a cliche', but if there's any way possible, play both. You might find that you prefer one over the other. I really like both. I don't know what the difference in price is (I got my Classic used at a great price). But if it is substantially cheaper, and money is an object, I don't think you'd be disappointed with the Classic. And most people are very happy with the "new" Am. Stds. ('08 >).