Furthermore and moreover are often understood as synonyms. However, there is a difference in meaning between them.

FURTHERMORE

Use furthermore when you add something to what you just said.

Earlier research has shown that this applies to several sectors. Furthermore, this has been confirmed in our interviews.

MOREOVER

Use moreover to indicate that you add something beyond what has been said, something different. Sometimes moreover could be said to mean ”further and more importantly”.

Using your cellphone while driving is against the law in some countries. Moreover, you risk your own life and that of others.

Sunset behind a town on a hilltop
The sunset was magnificent. Moreover, the whole day had been fantastic with walks in the narrow streets and a gorgeous lunch in the old town.

Furthermore and moreover are transitional words. Transitional words (or transition words) are used to describe relationships between ideas, to help the reader progress from one idea to the next. They can, for example, express addition (also, and, besides, further, likewise, again), contrast (but, however, on the contrary), time (after, before, usually, finally), space (above, below, behind, opposite), details (especially, particularly) and consequence (therefore, hence, consequently, because).

To sum up:
Furthermore (in addition to what has been said) adds information.

Moreover (beyond what has been said) builds up the argument, ”not only that”, adds a reason of a different kind, adds to diversity, ”more importantly”.

You can read about further and farther here.