When it comes to further as an adverb, the Longman Dictionary gives us the following explanations:
1. more, to a greater degree;
2. a greater distance, beyond a particular place; also farther (more can be found here);
3. time (into the past or the future);
4. used to introduce something additional that you want to talk about; also furthermore.
Furthermore is explained as “in addition to what has already been said”.
Putting aside the other meanings of "further", we can say that both words have the same meaning (“in addition to something”). They are both sentence adverbs, as well as formal. Also, from an etymological point of view, the adverb "furthermore" was created by binding together “further” and “more”. It seems like it hasn’t changed the meaning of “further” as “in addition to”.
Let’s take an example from Harry Potter:
He was not remotely pleased to see Fudge, whose occasional appearances, apart from being downright alarming in themselves, generally meant that he was about to hear some very bad news. Furthermore, Fudge was looking distinctly careworn. He was thinner, balder, and grayer, and his face had a crumpled look.
It is absolutely correct, in this example, to replace “furthermore” with “further”. The meaning of the sentence would not change in any way.
Based on experience, as well as the results of my current research for this article, I can say that “furthermore” is more often used than “further”. While they are both formal, “further” seems to be even more so. The only example I found was at the beginning of the book “A Thousand Splendid Suns” by Khaled Hosseini:
Further, the publisher does not have any control over and does not assume any responsibility for author or third-party websites or their content.
In this case as well, we can easily replace “further” with “furthermore”. However, judging by the lack of examples I have found with “further” as a sentence adverb, it seems that there is a tendency of using “furthermore” rather than “further.”
First, sentence adverbs can be either an adverb or an adverbial phrase “expressing the speaker’s attitude or classifying the discourse”. “Further” and “furthermore” are two examples of such sentence adverbs; others include: actually, apparently, basically, clearly, fortunately, hopefully, however, indeed, interestingly, ironically, naturally, presumably, seriously, thankfully, therefore, truthfully, and ultimately.
As sentence adverbs, both “further” and “furthermore” are quite often placed at the beginning of the sentence (followed by a comma). However, one must not assume that “further” placed at the beginning of a sentence is a sentence adverb, meaning “in addition to”.
Here are two examples from The Lord of the Rings series:
Further in his last battle Smaug destroyed the dwellings of the men of Esgaroth, and I am yet the servant of their Master.
Further and further abroad he gazed, until he cast his gaze upon Barad-dur.
Despite its position at the beginning of the sentence, in this case “further” cannot be replaced with “furthermore”; if it were, it would lead to a very confusing sentence. Therefore, it’s not the position of the word in the sentence that matters, but the meaning it carries.
In conclusion, “further” and “furthermore” are interchangeable when the former is an adverb (modifier of verbs, adjectives, other adverbs, phrases, or clauses; more here as well) and is used meaning “in addition to”, though “furthermore” is more common than its shorter form.