As I have seen lots of interest around this answer, more precisely around the number of legs (as @lfurini said), I have decided to open a question to find a canonical answer.

From the discussion (answer and comment section), one can see that the creatures are different, even though, as @jwenting pointed out:

for a long time the terms dragon and wyrm were used interchangably in English.

And from this answer one can find an effort to give standard definitions for them.

However, when it comes to specifics, more precisely the number of legs, there seems to exist lots of interpretations.

  • Do you have evidence that all these creatures exist in Middle-earth? Dragons and wyrms appear to but I don't think wyverns do. – TheLethalCarrot Nov 7 '19 at 9:41
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    @TheLethalCarrot "fellbeasts" are depicted as "wyverns". – Gonçalo Peres 龚燿禄 Nov 7 '19 at 9:49
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    In heraldry, the distinction between a dragon and a wyvern is that dragons have four legs while wyverns have two. Most modern fantasy keeps this distinction. – Klaus Æ. Mogensen Nov 7 '19 at 9:58
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    In Tolkien, you don't distinguish them. Glaurung and Smaug are referred to using dragon and worm/wyrm interchangeably. Wyvern isn't a word that Tolkien uses. – Nolimon Nov 7 '19 at 14:26
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    Dragons in Tolkien's mythology came in winged and wingless varieties, with the winged ones appearing (in-universe) at a later time. Wyrm (or worm, in Modern English) was the Anglo-Saxon word for dragon. I suspect the word "dragon" was only ever used, despite its Latin roots, because it would be widely understood what it meant. – chepner Nov 7 '19 at 21:50