My business is Dunlop tyres. My company is the Dunlop distributor in Malta and has been for over a century - father and grandfather before me, so perhaps my thoughts on such matters are biased.
Alot of thought is put into both the construction and compound of a racing tyre, and yes from my own experience I have found a noticible difference in handling even using the same construction but just changing compounds or vice versa.
Nowadays we have the added ingredient of cross ply or radial racing tyres too. In my racing days they were all cross ply.
We were running my son's single seater with Dunlop cross ply slicks for a while, we even tried a set of Avon A40 cross plys, but recently fitted a set of Dunlop's softest compound radials and the difference was chalk and cheese. The car was transformed and it is now absolutely glued to the track, with the resultant increase in sales of Dunlop racing tyres I'm happy to say. Suddenly everybody wants them!
What worries me with F1 at the moment is that a little variable like a slight change in tyre construction can suddenly change a car from being a perfect handler to being a pig. It's too delicate. his is where something needs to be done.
I will never understand for instance why Bridgestone, who now have a monopoly, still insist on producing a tyre which is marginal in certain conditions. Why don't they produce a tyre that will last, will not grain, will not overheat, will remain constant and linear in performance throughout a race. They can do it now since they don't have any competition, and so long as they are supplying everyone with the same tyres it wouldn't matter.
When the FIA announced that there was going to be just one manufacturer I thought, ah tyre problems will be a thing of the past. Unfortunately they are still one of the most critical items, which to my mind completely defeats the object of having a control tyre.