I began doing some research to better understand what I was doing when I make rough puff pastry. In the process, a neighbor who recently graduated from Le Cordon Bleu Paris explained that in France "dry butter" is always used for puff pastry but had no idea why. I found the definition of dry butter and the commercial product but not a good rationale for why it is important or even what it adds to the process or to the end product. One theory is that less water makes the butter melt at a higher temperature. I don't believe this to be true, though it may seem to have that effect. The thought experiment is to consider an emulsion of fat and water with varying amounts of water, and compare it to the pure fat. In this case you are comparing butter with various amounts of water (16% to 20% - the difference between buerre extra sec and standard US butter) to clarified butter or ghee (100% butter fat, available in South Asian markets). Ghee is quite hard and even at room temperature can be difficult to remove from a jar. At the same temperature standard US butter (whether salted or unsalted) is quite soft. In neither case is the butter melted. Butter is actually a mixture of low melting point fats and higher melting point fats and attempts have been made to re-balance the mixture toward low melting point fats to improve spreadability - with less than outstanding results.
I suspect that the advantage of "dry butter" comes from its physical properties as it is likely to have higher shear strength (than 80% butter fat butter) at cool room temperature and thus better match the texture of the dough into which it is to be laminated. Consider the case of crunchy bits of really dry butter (100% butterfat or clarified butter) which would be nearly impossible to laminate since it would not flow/creep and would create holes through sheets of dough unless that temperature was exactly right and would not have the water necessary to puff the dough. At the other end of the spectrum you have beaten butter that is spreadable, where the water has been released to combine with the flour and is then not available to form the steam so essential to putting the puff in puff pastry. In between is some emulsion of water in fat (I would suggest ~84% fat) where the shear strength is optimal for laminating a wheat flour based dough.
This is an entirely data-free analysis so please don't take it as fact, just as a candidate theory which might explain the benefit of dry butter in making puff pastry.