These are the Complete Demos for DOJO and ExtJS
DOJO http://archive.dojotoolkit.org/nightly/dojotoolkit/dijit/themes/themeTester.html
ExtJS http://dev.sencha.com/deploy/dev/examples/themes/index.html
Which one of them is better - Aesthetically pleasing and performant both ? Also which of them would be a better Candidate for Webkit based Desktop/Mobile Application ?
Aesthetics are subjective, and not quantifiable. I personally prefer the look of ExtJS, but thats me. I'm sure others prefer the look of Dojo.
Dojo is free, ExtJS is commercial (unless its an in-house app).
ExtJS has a pretty awesome API, once you get used to how it thinks. Its very object oriented, and has a great plugin/extension eco system.
I can't comment on performance, never bench marked them.
The requirement of which to use would come down to the project requirements. Both can be themed, so default aesthetics may not be as important as the bredth of functionality they offer. The ExtJS grids are exceptionally powerful, so if you're doing a lot of grid based data work that might be a plus in Ext's direction.
Compre their features, and align that to your needs. If no clear winner presents its self, and money isn't an option just choose the one you feel most productive working with.