Sigma 10-20mm F3.5 vs F4-5.6

Does the newer F3.5 version of this lens have any advantages over the older F4-5.6 version?

From the specs slighty faster and the same aperture at all focal lengths.

Yes, but the question is do you notice the difference in practice? And if the answer is yes, is it worth the extra money or not? I'm considering buying one of these lenses, but I'm not really sure which one I should get.

I used to own the Sigma 10-20mm f3.5 version. If you are going to shoot in low light or night photography then the f3.5 is handy as you can hold the aperture open through the whole focal range length range. If you are using it in an underwater housing be aware that the f3.5 version is slightly larger than the f4-6.5 version.

I'd go with the older f/4-5.6 version for underwater. Save cash, it's smaller and lighter (like Grant said), and you'll likely not use it wide open (at f/3.5) underwater. But, if starscapes and indoor shooting interest you, then consider the f/3.5.