Many is when there is a definite number. For example, I went to the market to buy tomatoes and the seller asked me: How many tomatoes do you want? I said : I want three tomatoes.

Much is used when there isn't a definite value. For example, my mother made my favorite tomato soup and asked me: How much do you want? I said : As much as it takes to fill up the bowl

  I have much difficulty speaking a foreign language.

  There is much to think about. There is much work to do.

There is not much that we can do about this issue.

  I like you very much.  I like watching Formula One  racing very much.

There is way too much emotion   involved with this discussion.

  I just don't care very much  for that kind of career.

Much of the difficult work has been left for us to complete.

I do not think that this is helping very much.

  This is not much of a complete or thorough address of the subject, but I hope it helps.

I just do not like that teacher very much.

There is too much coldness and indifference  in his attitude.

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   I have many interests.

I have many friends but also many adversaries.

Many people become   entirely to emotional   about these discussions.

Many people  can do this so much better than I can.

I can give you many examples.

There aren't many cookies left..

Too many people  offer personal opinions rather than facts.

Many people fail to show  kindness and courtesy.

I think many students of English will benefit from these examples.

I think only a few students of English will actually benefit from these many examples.

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"Much" is used when something cannot be counted such as milk, money, etc...

Examples:
How much milk did you drink?
We don't have much gas left.

"Many" is used when something can be counted. You can say, : "2 people", "3 apples", or "4 cartons of milk".

Examples:
Many people came for the unusual event in the park.
Today, many schools offer computer courses for their students.
How many apples did you pick?
She bought three cartons of milk.

Yes, as the video and PDF explain, 'much' is used with uncountable nouns and 'many' is used with countable nouns.

Even as a native speaker of English I did not know that!