Many modifies things that can be counted (i.e., count nouns). Much modifies things that can’t be counted (i.e., mass nouns). In other words, many tends to modify plural nouns, and much tends to modify singular nouns. For example, we write many doctors, many stars, and many dollars because these nouns are countable, and we write much knowledge, much light, and much money because these nouns aren’t countable (in their main senses).
There is little controversy around the many-much distinction, and it is borne out with only rare exceptions. You might occasionally encounter a much dollars or a much people, but few English speakers use such constructions out of habit.
The distinction can be tricky in a few rare instances. For example, while many thanks is more common than much thanks, the latter appears fairly often because we can think of this thanks as a mass noun (synonymous with gratitude) that takes a plural form by convention. Of course we can also think of thanks as an abbreviation of the plural noun thank-yous, in which case many is appropriate. And then there are singular mass nouns that sound plural. Kudos, for one, is a mass noun that happens to end in s, so the phrase many kudos is more common (on the web) than much kudos even though the latter is more logical.
Say to her, we have measur’d many miles / To tread a measure with her on this grass. [Love’s Labour’s Lost, William Shakespeare]
In the nick of time, and amid much political drama, Congress passed the American Taxpayer Relief Act on New Year’s Day. [Wall Street Journal]
The interview raised many more questions that many observers will hope will be answered. [Sydney Morning Herald]
I am larger, better than I thought; / I did not know I held so much goodness. [“Song of the Open Road,” Walt Whitman]
Many secondary mechanisms regulate the evolution of damage and repair after traumatic brain injury. [“The Many Roles of Adenosine in Traumatic Brain Injury”]
Much fuss has been made about the vitriolic chemistry between Mariah Carey and Nicki Minaj. [WetPaint]