Is Spaetzle a pasta or a dumpling?

Spätzle originates from a region spanning Austria, Switzerland and southern Germany. I can tell you that here in southern Germany, Spätzle is definitely considered a sort of pasta. You can buy premade, prepackaged dried Spätzle, and it is always on the pasta shelf. So I'd say it is correct to refer to it as pasta, at least for historical/traditional reasons, even if it doesn't meet some technical criteria for pasta (or does meet some for dumplings).

As for technical criteria, I don't know of any decision criteria for either category both are loosely defined in my head by enumeration of their elements. Maybe there are some official criteria, but kitchen taxonomy isn't as hard defined as e.g. biological taxonomy, and it varies by region/country. Consider for example, the German word Braten, which is generally translated as a roast. But while a German cook may consider a piece of meat cooked in a dutch oven on a stovetop a kind of Braten, any cook from the Balkans will tell you that this is not a roast, as it wasn't made in a "real" oven, but in what is, in his eyes, a pot. So would you define the meat as a roast or not? I'd say that here regional tradition is best. If it was a German recipe, you are free to call it a roast. If it was a Balkan recipe, you'll have to call it something else, depending on the exact recipe. Similarly, for Spätzle, I'd say just go with the tradition and call it a pasta.

I would say Spätzle is closer to pasta than a dumpling - it's a dough similar to pasta (eggs, flour, water, salt).

For me, the main difference between pasta and dumplings is that pasta is cooked in water and sauce is added later, while dumplings are often cooked and served in the broth which flavours them.

The English language wikipedia page http://en.wikipedia.org/wiki/Spaetzle also seems to suggest that it's an adaptation of pasta, especially since it's associated with those areas of Germany, Hungary, Austria etc which are closer to Italy.

There isn't really a difference between them, or at least it is a very loose one, since dumplings can be regarded as a variety of pasta, such as tortellini.
In Italian language, "pasta" commonly indicates just dried durum-wheat pasta or fresh egg pasta (such as spaghetti, maccheroni, etc.) while there isn't an exact translation for "dumpling"; stuffed pasta like ravioli, tortellini and cannelloni AFAIK lacks a collective name and just go by their own name.

i would say the difference between pasta and dumpling is the shape. dumplings are round, ball-like shaped, like quenelles, and pasta is the pasta thing we all know. with a spaetzle-dough you can make quenelles as well as long thin pasta.in parts of germany and austria they make spaetzle long and thin, scratching the dough from a wooden board with the blunt side of a knive into boiling salted water, and in other parts they make little quenelles, scratching the dough through a special spaetzle- sieve into boiling salted water.