We do have different words for them, yes (devil will be "djævel", and demon translates into "dæmon", so they are actually rather close to their English equivalents (and just to mention it because it seems interesting, the English word "imp" would be translated into "smådjævel" - literally "little devil"), and it is, indeed, the same word we use about the Christian devil (though we usually make a distinction simply by adding a definite grammatical article to the latter - "djævelen"). That said, we do have certain rather glaring difficulties when it comes to connecting Danish terms to particular mythical creatures, since each mythical creature usually have a number of different names, and different creature-names can typically be used about any number of different creatures. Leprechaun, for instance, is really hard to find a Danish word for... we'll call them "trolde" (trolls) or "nisser" (I guess... elves? Gnomes? It's a word that refers to a lot of small mythical creatures). Reading Tolkien's novels is also a rather different experience in Danish, by the way, since a very wide array of different creatures in it - ogres (I'm not even sure ogres are in there, but I mention them all the same because they are on the list), trolls, orcs, goblins and hobgoblins (Uruk-Hai or something similar in Lord of the Rings) are, in Danish, all simply "trolde"... trolls. The distinction is made by linking to their usual habitat instead (goblins are cave trolls, orcs are mountain trolls (though often referred to as "orker"), trolls are forest trolls, and so on and so forth... collectively they are trolls, though).
Trolls are [I]very[/I] common in Danish folklore, and are actually often surprisingly human in their behavior compared to the otherwise more commonly recognized as human-term, witches. Witches can just as easily be monsters as they can humans in our folklore, too, but the common trait for all witches is that they are uncompromisingly evil... which doesn't really make sense, since when we look at our [I]real[/I] witches, and those featured in Norse mythology, they've almost exclusively been good entities. We have a lot of much more obscure creatures, too, a lot of which are actually somewhat interesting to compare to more widely known entities. (Danish elves, for example, have always been [I]evil[/I] and enjoyed luring people (usually men) to their death in various gruesome ways, often by seducing them in one way or another. The seduction is very rarely with any sexual nature, surprisingly, but usually belies on tempting them into somehow betraying the way of life that was commonly acceptable at the time.)
Or lindworms (or lindwurms or however you want to spell it; in Danish it's "lindorm"). As the name suggests they have always been viewed as snakes here, albeit sometimes with legs. There's never been a lindworm with wings, though, and they always breathe lethal toxin. Some refer to Jörmungandr as the greatest of the lindworms, too. (I could never get used to call it Jörmungandr, somehow... in Danish we usually refer to it as an approximate translation of its other name, the Midgard Serpent (Midgårdsormen).)