In what way did Shakespeare create a distinction between the educated nobility and the coarse lower classes?

This is a very simplistic question because the distinction was clearly maintained in real life and that was only carried forward into Shakespeare's plays. The most obvious difference between people of different social classes was their clothes. People were forbidden by law to dress in certain ways unless they were rich and noble enough. The costumes used in the plays showed this: the actors playing noble people wore fine clothing (the castoffs of the real nobility).

The other difference between the upper and lower class people is the way they talk. Shakespeare often puts a stately blank verse in the mouths of the upper crust and arrhythmic prose in the mouths of the common people. But not always. Even the nobility speak in prose when they are disturbed or insane, and they speak in prose all the way through Much Ado About Nothing. Prince Hal talks in prose when talking to Ned Poins. Blank verse is saved for matters of seriousness where a more poetic approach is needed. It is not, therefore, a matter of social class so much as a matter of the weightiness of what is being said (and in Shakespeare, the lower classes rarely have anything worthwhile to say).