There aren’t many; when a set appears, one or the other of the words tends to drop out of use or take on new meaning.
I submit:
unmercifully/mercilessly
What would be the point in a language that contained extraneous words that meant the exact same thing? At that point, wouldn’t one or the other of the words take on a separate connotation that would distinguish it from the other word? Would one cease to exist, or become archaic?
There’s quite a host of questions about the evolution of language here.
You mean like flamable / inflamable ?
There you go. Except “inflammable” carries the modern, “incorrect” meaning of “not burnable”, which is why it’s not used as a warning anymore when something IS burnable. A shade of nuance between the words. I’m looking for pairs with no shades of nuance; they mean exactly the same thing to nearly everyone.
drawing pin / thumb tack
completely interchangeable where I live.
OTTOMH:
frightened/scared
moisten/dampen
eyeglasses/spectacles
footpath/sidewalk
rise / arise
iterate/reiterate
alogical/illogical
the meaning of a word is relative to and dependent on the rest of the vocabulary and grammar of that language, so no word has a fixed, autonomous and unambiguous “meaning.”
so for two words to have the same exact meaning is not possible – as no one word has an exact meaning.
the pair of words in the OP are not synonyms, they are two inflectional forms of the same word.
guess i dropped a wrecking ball on this thread … 
Don’t worry about it in the least; I’m going for a record for lowest thread/post ratio, or Whose Threads Die Fastest. 
I don’t know the word “alogical”. Is that a real one?
I’m trying to do my best to help that ratio along! 
When I think of most of these pairs of words, I get a slightly different general impression from each word. Then again, maybe I’m defining words incorrectly. (For example, moisten/dampen: I think of something moistened as being wetter than something dampened.)
Yes, there’s kind of a continuum, isn’t there -
dampen - moisten - wet - saturate - soak - enliquidinate -
What?