They are synonyms and they aren´t.
Depending on the effect someone mean, they have pretty different usages.
Besides adverb, preposition, according to the Cambridge Dictionary means: in addition to; also
Moreover adverb - formal (used to add information) also and more importantly
The whole report is badly written. Moreover, it's inaccurate.
Besides (examples from the above said dictionary:
Do you play any other sports besides football and basketball? (1)
She won't mind your being late - besides, it's hardly your fault. (2)
In (2) it can be replace by besides, and why?
Because in (2) the word in question gives further or more specific information on, expands in some way the information from the first part of the sentence.
Notice that both words (moreover/besides) do that, so in (2) they are equivalent, and would normally be preceded by a comma, hyphen (for lack of a better word here, I forgot the term now...), something that pauses somehow the sentence, isn´t part of it (of the sentence, I mean).
"besides" could work as a "concessive linking adjunct" to indicate that the writer accepts part of an argument or proposition, but in general, he is against the whole idea or his idea somehow contrast with the previous one.
Generally, in this way, words like "besides/anyhow/nevertheless" come in the second segment of the sentence. Sometimes between commas - other times in a separate-related statement, and somehow contrasted with what was stated before.
I mean, they can´t sit and be quiet about it, as if it doesn´t exist. That´s my opinion, anyhow.
In the case of "besides" it may add another reason to an action, so it´s somehow about concession-contrast. One concedes a point, but then give another reason why something happened (in Portuguese could be translated into "além disso":
Barbara majored in biology because she was fascinated by the subject; besides, she knew it would help her get a high-paying job in the future.
Sources: sites and the Cambridge Grammar of English - a Comprehensive Guide, with some tinkering on my part.