You can read the definitions of the two words and you may find, though they use different words to describe them, they seem to be talking about the same thing. One might lead to the other (as an inference), but that is not necessary and native speakers would not confuse these for each other. The primary difference is that:
'annoying' is something that is actively bothering you. A loud noise, or something that keeps interrupting you.
'boring' is something that is dull and lifeless, or at least seems so to you.
Both are things that you'd rather stop, but you wouldn't mistake one for the other, and so you wouldn't consider them synonyms. Of course something that is boring may well be annoying you (because it is so boring), but that doesn't make the two concepts the same.
Note that even if the lists of synonyms for each overlap, that doesn't guarantee that they are synonymous. (in mathematical terms, synonymy sounds like it should be an 'equivalence relation' but is not. 'synonymy' is more like 'near' rather than 'equal to'.