Ljubljana, Slovenia.
Temperature: -3°C Clouds: Broken Clouds
Maribor, Slovenia.
Temperature: 2°C Clouds: Overcast
Portoroz, Slovenia.
Temperature: 1°C Conditions: Mist

Top GIS results for "Slovenes" (left) and "Slovenians." (right)
In 2001, a team of linguists at the Slovenian Academy of Sciences and Arts, led by Jože ToporiÅ¡iÄ?, unleashed a 2.000-page beast named Slovenski pravopis (Slovenian Orthography) on the world. That book is the authoritative source on the Slovenian language, painstakingly documenting all its complex spelling and grammar rules.
English ain’t so lucky. There is no single source to turn to for answers, which makes it slightly more difficult to answer the question: "Are people from Slovenia called Slovenes or Slovenians?" But we can still try. And we will!
The Short answer: Both are acceptable. Neither one is wrong. Consistency is always nice.
The Long answer: All of the above, but "Slovene" is the first choice among authoritative sources. Here’s a quick breakdown of how some of the most reputable authorities approach the problem:
The Oxford English Dictionary
Description: The "definitive record of the English language."
Decision: Lists "Slovene" first (as person or language) and "Slovenian" as a derivative (noun or adjective.) Other dictionaries seem to follow this convention: Both Merriam-Webster and the American Heritage Dictionary list "Slovene" as their primary entry.
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The New York Times
Description: "The paper of record."
Decision: Uses "Slovene" as a noun, and "Slovenian" as an adjective. (Note: This is the rule I try to follow.)
(Source)
The British Broadcasting Corporation (BBC)
Description: "The largest broadcast news gathering operation in the world."
Decision: Favors "Slovene." In fact, there is a subdivision of the BBC called BBC Slovene.
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The Government of the Republic of Slovenia
Description: The folks running this country.
Decision: Use both interchangeably, but overall seem to favor "Slovenes" (313 hits) over "Slovenians" (131). The official Facts About Slovenia booklet, published last month by the government’s Public Relations and Media Office, exclusively uses "Slovene" as a noun and "Slovenian" as an adjective. You can read that 116-page booklet here. (PDF)
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Some Other Sources
The Central Intelligence Agency (CIA)
Description: "U.S. agency responsible for obtaining and analyzing information about foreign governments"
Decision: "Slovene" as noun, "Slovenian" as adjective.
(Source)
Google
Description: Index of 8,058,044,651 web pages.
Decision: Favors Slovene. Currently lists 73,800 hits for "Slovenes" and 42,900 hits for "Slovenians."
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Slovenian Chamber of Commerce
Description: Association of businesses in Slovenia.
Decision: Uses "Slovenian" exclusively.
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Matkurja
Description: Online directory of Slovenian webpages and country guide.
Decision: Mixed. Sometimes seems to follow the "Slovene" as noun, "Slovenian" as adjective rule.
(Source)
This could go on forever, but it basically boils down to what sources you respect and your personal preference. The big four listed at the top (OED, NYT, BBC, Slovenian Government) are good enough for me, and basically trump the fact that, say, the Football Association of Slovenia uses "Slovenian" exclusively. So I’ll keep on truckin’ with "Slovene" as noun and "Slovenian" as adjective.
At any rate, this will probably be the last time I address the subject, so I welcome any and all comments!