Safest to use a lot of, as people have commented above it can be used with countable and uncountable nouns. Many is best used with another word like "too" when making a positive statement There were a lot of people on the bus. There were too many people on the bus. It is ok in questions and negative statements: How many people were on the bus? There weren't many people on the bus.

Many can be used only for count nouns. eg. many tables, many shirts etc. But 'a lot of' can be used for count and non-count nouns both. eg. a lot of tables, a lot of water etc.

The only difference I can think of is 'many' is more formal. Other than that they mean exactly the same thing.

Ok, if you need extra information you could use "Much" instead of "many" when you refer to uncountable nouns..

Joel Rengifo, thank you for your help! I will think about countable and uncountable when I don't know whether I should use 'many.'