lot of or lots of is usually used in speaking English, so it is a informal word to express something. Also, lot of or lots of can be followed a n （U） such as, lot of water/money Many is used with plural nouns and verbs, especially in negative sentences or in more formal English, to mean "a large number of",Also, used in questions to ask about the size of a number, and with "as/so/too". such as, many people/students , How many children do you have? usually, many can be used in both speaking or writing English.

Many is used only with countable nouns. It is used mainly in questions and negative sentences: Do you go to many concerts? ◆ How many people came to the meeting? ◆ I don't go to many concerts. Although it is not common in statements, it is used after so, as and too: You made too many mistakes. In statements a lot of or lots of (informal) are much more common: I go to a lot of concerts. ◆ 'How many CDs have you got?' 'Lots!' However, they are not used with measurements of time or distance: I stayed in England many/quite a few/ten weeks. A lot of/lots of is still felt to be informal, especially in BrE, so in formal writing it is better to use many or a large number of in statements. (from Oxford Dictionary)

They are both similar - "many" is a bit stronger and implies a larger number than "a lot of". e.g "There are a lot of people here" or "there are many people here". We tend to use "a lot of" in ordinary speech, rather than many. In some sentences "many" just doesn't sound right. For example, "I have a lot of work" not "I have many work".

"a lot of" можно использовать с любыми существительными. "many" - только с исчисляемыми, а с неисчисляемыми будет "much".