(many) thanks

There isn't a big difference. "Many thanks" is another, slightly more informal way of saying "Thank you very much."

thanks (noun)
1. an acknowledgment of appreciation
2. with the help of or owing to; "thanks to hard work it was a great success"

http://dictionary.reference.com/search?q=thanks&r=66

Many thanks:

An expression of gratitude, expressing greater gratitude than thank you or thanks.

http://en.wiktionary.org/wiki/Many_thanks

As Shapeless says, it's simply less formal.