We did a back to back test at Mt Tremblant last weekend in the Lotus 69 - in the same session we changed from a good set of Dunlops to a new set of Avons. The best lap time was 0.3 second quicker on the Avons but I would be hard pressed to tell the difference between the two tires. The Avons seemed to have slightly more grip going through turn 2 which as anyone who has been there knows is very quick indeed. The Dunlops seemed to have a tad better braking feel going into the hairpin. Both tires give great feedback when really pushing the car (which I was - below 1.50 laps) but maybe the Dunlops give slightly more feedback when the car gets sideways (Dave Handy tells me the Avons may have slightly stiffer sidewalls).
To summarize; Avon have done a great job in 'copying' the Dunlops and I am convinced there is no noticeable perfomance advantage from either brand.