In meaning, no. Both mean "Additionally" but you can't always replace one with the other. e.g.: Nico stopped running because he was tired. Besides, he hadn't wanted to run anyways. "Besides" is informal and "furthermore" is formal. In this example, I think that is the reason why "furthermore" doesn't sound right. e.g.: Besides being detrimental to human health, smoking causes one in ten forest fires. Here, you cannot replace "besides" with "furthermore" because you can't attach a phrase to "furthermore". To use furthermore you would have to split the sentence into two: Smoking is detrimental to human health. Furthermore, it is harmful to the environment: cigarettes cause one in ten forest fires.

"Furthermore" means "in addition". However, this is a very formal word and is often only used in written English, for example, when I write an essay, I would write "furthermore" instead of "as well" as it sounds more formal. People don't generally say this word. "Besides" means " "apart from". An example sentence is: Besides (Apart from) the Scottish, the people of the United Kingdom want to remain a part of the United Kingdom.

There is no difference. The are both used when you are adding another stronger reason to support what you are saying.