In French, some older books distinguish between l’Antarctide, the continent, and l’Océan Antarctique, the surrounding ocean. I think it fell out of favour around the 1960s; now everybody says l’Antarctique for the whole South pole. Since it referred to a continent, and there’s no continent under the North pole, Arctide is unheard of.
I blame global warming and the melting of the ice shelves in Antarctis.
“Antártida
Español
Pronunciación (AFI): [an̪.ˈtar.ti.ða]
Etimología
De antártico, a su vez del griego ανταρκτικως (antarktikos) ‘opuesto a ártico’, y la terminación -da para asimilarlo a otros nombres geográficos.”
From antártico, which is from Greek antarktikos, ‘opposite the Arctic’, and the ending -da is to make it resemble other geographic names.
Spanish Wikipedia also notes that Chile differs by calling it “Antártica.”
Cite: RAE (2005). «Diccionario panhispánico de dudas - Antártida». lema.rae.es. Consultado el 15 de febrero de 2013. «Para denominar el conjunto de tierras situado en el polo sur terrestre son válidas las denominaciones Antártida y Antártica. La forma etimológica Antártica […] es de uso mayoritario en Chile […]. La forma Antártida […] es la única usada en España y la preferida en la mayor parte de América».
The Pan-Hispanic Dictionary of Doubts says: To name the landmass located at the South Pole, both names *Antártida *and Antártica are valid. The etymological form *Antártica *is used by the majority in Chile. The form *Antártida *is the only one used in Spain and is the preferred one in the greater part of the Americas.
The Academia seems to imply that Antártica is the form backed up by etymology, while Antártida was made up just so that it would sound more like other places. I.e. the Academia just thought it sounded better that way. And they admit it’s a “doubt.”
Like América, África, Asia, and Australia, the names of the other continents in Spanish. 
Hey, I wish I knew what they were talking about too! Apart from Atlántida for Atlantis, nothing comes to mind.
So I went and looked up other Ancient Greek toponyms ending in iota sigma like Atlantis. And sure enough—
Aeolis > Eólida
Argolis > Argólide
Colchis > Colquida
Elis > Élide
Locris > Lócride
Phocis > Fócida
Phthiotis > Ftiótide
Propontis > Propóntida
Not sure how they decide when to use -ida and when -ide. The Ancient Greek declensions of iota sigma words that I looked up have -idos in the genitive and -ida in the accusative. Couldn’t tell where exactly the -ide variant came from.
Anyway, that sure answers the question of why the Real Academia chose to use an inflected Ancient Greek ending! Because it’s Ancient Greek, ¡claro!
And I would not be surprised if Chile uses “Antártica” just to be contrary to the Argentines. (The Chilean and Argentine claims in Antarctica overlap each other)
They tend to have both variants, with both being valid; I know I’ve heard both Cólquida and Cólquide, Argólida and Argólide. Which one got added to the dictionary or to a map, or used in a text, generally boils down to “which version did the person doing the work think of”. It can also depend on what comes right after: if the next word begins by a-, use the variant which ends in -e.
Indeed, there is a chill in the relations between those countries.
I just broke in to say that “Arctic” as in “Arctic Ocean” is not an adjective. No more than “North” in “North Pole”. Sometimes it is not clear whether a word modifying a noun is an adjective or a noun used attributively. But your internal grammar knows even if you don’t. Do you say Arctic Ocean or Arctic Ocean? The former, right. That marks an attributive. You might say the latter but only if you were contrasting the Ocean with something else, Arctic ice, maybe.
The following is a true statement, as it happens, that illustrates the difference. There is a brown building on the McGill campus called the Brown building.
That’s fine, I’ll start pronouncing it that way when the French quit calling London Londres.