Difference in cookie texture if we use melted vs softened butter
There are many factors in play such as the type of sugars, amount of eggs or other sources of hydration, amount and type of leavening and so forth, but as an overall generalization:


Melting the butter will lead to chewier cookies
Creaming colder/room temperature butter with sugar will lead to cookies with a higher, more cake like texture.
Refrigerating the dough before baking will help inhibit spread because the butter is colder, and takes longer to melt.  It is also gives time for the liquid in the recipe to hydrate the flour.   See What does an overnight chill do to cookie dough, that a 4 hour chill doesn&#39;t?


See the transcript of Alton Brown's Three Chips for Sister Marsha episode for a good treatment of chocolate chip cookies and their variations.