Both of them can be approximately translated as "Many",or 很多 can be translated as "a great many", "a lot of "... but the difference is : 许多 simply describes the "quantity", while "很多" （很 means "very"）lays more emphasis on the "extent", has the potential meaning of comparing to the normal situation e.g. 他有许多书＝he has many books 他有很多书＝he has a great many books (potentially means "more than normal people have")

许多 Many 很多

许多 and 很多 is the same meaning

Doesn't matter too much. 许多is more formal and more often used in literature than很多

for me no many difference, almost the same. "许多" is used by children more, I think. It sounds cuter.