Difference between 许多 and 很多
"很多" is not a word, but an adverb "很" and an adjective "多" combined to form a subordinate phrase. "很" as an adverb, is generally placed before an adjective to indicate a high degree of something. Therefore, "很多" has no particular significance. "很 + adjective" when used to modify the noun generally contains "的", such as:


很好的计划 (a very good plan)
很可爱的孩子 (very cute children)


But "很多", when used to modify the noun, is generally without "的". Such as:


很多人 (many people)
很多地方 (many places)
很多东西 (a lot of things)
很多城市 (many cities)


"许多" is a numeral (数词), meaning in great quantities. "很多" can be disassembled; "许多" cannot. The usage of "很多" is more flexible, and can serve as a predicative, attributive, or complementary component in a sentence. Such as:


南京路上人很多（谓语） On Nanjing Road, there are a lot of people
(predicate)
我去过很多地方（定语） I have been to many places (attributive)
有些人吃得很多，却不爱运动（补语） Some people eat a lot, but do not exercise
(complement)


"许多", as a numeral, can only be used as an attributive adjunct which generally modifies the noun. Such as:


许多优秀人才 (a lot of talented people)
中国的许多城市 (many cities in China)
那火山已经沉睡许多年 (that volcano has been laying dormant for many years)
许多消费者都不满意 (many consumers are not satisfied)
许多人在生活中离不开手机 (many people cannot live without their mobile phones)


The following statements are wrong when used with "许多":


  南京路上人许多（谓语）X 有些人吃得许多，却不爱运动（补语）X


In the dialogue, "很多" can be used alone, but not "许多". Such as:


  排队的人多吗？——很多。√ (Is there a lot of people at the queue? - A lot.
  √ ) 排队的人多吗？——许多。X


To answer your question, "许多" is more formal than "很多", and is used more often in writing, whereas "很多" is used often in speech. "许多" can be replaced by "很多", but usually not the other way round:


  过了许多（or 很多）年的教学生涯，我空闲时缅怀的往事很多。



Adapted from: “很多”与“许多” - 复旦大学国际文化交流学院