"Many" and "much" both express the idea of a great quantity, or a great number of things or people. "Much" is used with uncountable items (general things that you cannot separate/divide, such as "water" or "noise" or "joy") – "there was much water soaking the ground after the rain", "there was much joy in her voice", etc.
Many is used with countable items, i.e. when there is more than one item, as in "there are many people in the street", "I have many things to do", "I have many miles to walk". So you can’t say "there’s many distance between my house and my friend’s house". "Distance" is a singular, with quite a vague meaning here, so the proper phrasing should be "there’s much distance". It is traditionally considered that "many" and "much" express a subjective perception of number or quantity, while "a lot of/lots of" (which works both for countable and uncountable nouns) is felt to be more objective/technical.
Other words can express this very idea of "a great number of": "several" (adj.) : very neutral and formal, "numerous" (adj.) : a literary word, "a large number of" : neutral and formal, "diverse" + noun : idea of variety, as well as other, more familiar words such as "plenty of", "a great deal of" : the latter two can be used with countable and uncountable nouns alike.
LAST BUT NOT LEAST, the expression "many a + noun", although formal and literary, can be used to express the idea of repetition, of something that keeps happening. Example: "Many a time, I made the same mistake" (time after time, over and over again, I kept repeating it…).
