I think, more often we should use "many" with countable nouns while "much" with non-countable: "too much oxygen" (can not be interchanged with "many") "much rain" "too much laughters" etc. "many cars" "many problems" "many times" "a lot" is synonymous to "much" "I feel a lot better" = "I feel much better." :)

"I do not like this much" (meaning I like this very little) "I go to the park a lot" (meaning I go often) "There are lots of apples" (meaning many) These are only a couple of examples. Many teachers in the US say that we should not use a lot or lots, but they are still common.