there really is no difference, they both mean the same
thing.
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According to "Obscure Words," by Michael A. Fischer (ironically labeled by their dedicated compiler as "WWFTDs" - Worthless Words for the Day - "sesquipedalophobia" means the "fear of long words." Go figure!
And an anecdote
Another obscure word that means the same is "hippopotomonstrosousquipedaliaphobia."
See the Related link.
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Hippopotomonstrosesquippedaliophobia means the fear of big words, so he would be a hippopotomonstrosesquippedaliophobe.
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Both the words Logophobia and Verbophobia mean the fear of words. Onomatophobia is the fear of hearing a certain word. Sesquipedalophobia and Hippopotomonstrosesquipedalipophobia both mean the fear of long words.
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Pneumono-ultramicroscopic-silicovolcano-coniosiphobia. Even I'm not sure what it is.