Just from a look at the specs on Newegg, it looks like the "Extreme Core" edition has a slightly higher overclock. The regular Amp edition has a 1.569 GHz core clock and 1.683 GHz boost, while the Extreme Core is clocked at 1.607 core and 1.721 boost. That works out to a little over 2% difference in clock rates between the two, and probably not more than a 2% difference in actual performance.
They also have a somewhat different cooler. The regular Amp's cooler has two 100mm fans, while the Extreme Core's has three 90mm fans, which might potentially help provide better cooling. It also appears to have some LED lighting.
There also appears to be another one, called "Extreme" (not "Extreme Core") that has even higher clocks, 1.645 core, 1.759 boost, which is clocked about another 2% higher still, with the same cooler as the Extreme Core, but it's out of stock at Newegg.
The Arctic Storm appears to be equipped with a water block to connect with water cooling setups, but it's clocked lower than all of them, at least by default. Given adequate cooling, all these cards could probably overclock to similar levels though, so the main difference is probably the cooler they come with.