Students of Japanese may have at some point wondered, as have I, what the difference between ないです and ありません is. Ask a Japanese, and you will get a vague explanation about how the former is less formal than the latter, and how there is no difference in their meaning. That explanation is valid—but it is not the entire truth.
The difference between the two is precisely the same as the difference between It’s me and It is I: the former is incorrect according to the rules of traditional grammar, but it is often preferred because the latter sounds stiff.
The word ない (無い) is an adjective (形容詞).[[These 形容詞 are often called ‘i-adjectives’ or ‘adjectival verbs’ to distinguish them from 形容動詞 (‘na-adjectives’ or ‘adjectival nouns’), but for simplicity’s sake, and reasons given here, I will just call them ‘adjectives’.]] です is a heavily contracted combination of the particles にて, and probably either 候ふ (sōrō) or ございます.
です means essentially the same thing as である, だ, and でございます. Each is a combination of the particle で (contracted にて) and a verb that means ‘is’. です can be exchanged for だ, である, です, であります, ございます, or で候ふ. The phrases are semantically and grammatically identical: replacing one with another changes a sentence’s tone but not its meaning.
However, unlike the others, です is used after adjectives. The acceptance of this usage dates back to 1952, when it was formally approved by the National Diet.
７ 形容詞と「です」
これからの敬語
これまで久しく問題となっていた形容詞の結び方――たとえば，「大きいです」「小さいです」などは，平明・簡素な形として認めてよい。
7 Adjectives and ‘です’
The long-controversial use of [of です] with an adjective to end a sentence—as in ‘大きいです’, ‘小さいです’, etc.—is acceptable as a plain and simple form.
The adjective+です construction is accepted by Daijirin.
… 形容詞および形容詞型活用の助動詞には，その終止形に付く。
Daijirin, entry for ‘です’
… when adjectives and adjectival auxiliary verbs are used, [です is] attached to their terminal form.
And it has been accepted by Kōjien since 1998.
「面白いです」のような形容詞に付いた言い方は、昭和１０年代までは由緒のないものとされたが、現代は正しいものと認められている。
Kōjien 6th ed. (2008), entry for ‘です’
Until the second decade of the Showa period[[This means around 1935–1945.]], expressions such as ‘面白いです’, in which です is attached to an adjective, were considered unorthodox; but today[[The sixth edition was published ten years later in 2008, but the entry is the same as that in the fifth.]] they are considered to be correct.
The fourth edition of the Kōjien, in which Shinmura Izuru’s influence is more evident, does not even mention the use of adjectives after です:
【助動】(「で候(そう)」の約とか、「でござります」の転とかいう) 体言や体言に準ずるもの、或る種の助詞に付けて、指定の意を表す。[[①狂言では、おもに大名・山伏などの名乗りなどに使い、尊大な感じを表す。である。狂、禰宜山伏「これは出羽の羽黒山より出たる駈出の山伏です」
②江戸後期に、花柳界の人、医者、職人などの語。人、春色江戸紫「マア何時頃でせう」。人、春色玉襷「さうでしたか、あんまりおまはんが嫌がつて逃出すものだから」。滑、人心覗機関(のぞきからくり)「医者か飛脚か我ながらげせぬです」。人、春色江戸紫「身につまさるるやうですねへ」
→でえす
→でえんす。
③幕末近くになると、普通の商家の婦人も使うようになり、明治時代以後、特殊社会の語であることが忘れられて一般に使われ、「だ」に対し、「でございます」よりは軽い丁寧な語になった。広辞苑第四版（1991年）]]
Kōjien 4th ed. (1991)
【Auxiliary Verb】(Said to be a contraction of ‘で候 (そう)’ or an alteration of ‘でございます’, or similar.) [です is] attached to uninflectable or nominalised words, and to some types of particles, and has the connotations specified [below].[[The connotations specified are (1) in a Noh farce, the haughtiness of of a daimyō or a mountain hermit-monk; (2) in the late Edo period, the speech of the pleasure quarters, and of physicians, artisans, etc.; and (3) from around the Meiji Restoration onwards, a level of politeness above that of ‘だ’ and below that of ‘でございます’.]]
Foreign writers of Japanese may sometimes wish to eschew the adjective+です construction to avoid sounding foreign or childish. Although it is widely accepted as proper Japanese, it is not considered formal or sophisticated.
Although we can rewrite 面白くないです as 面白くありません, we cannot rewrite 面白いです as 面白くあります. It is not technically wrong, but it is not conventional Japanese at all, and will cause the reader to stumble. The phrase 面白くないのです is grammatically correct (I believe の is short for もの), but is has an explanatory tone that is not always appropriate. 面白い is correct on its own, but is not formal. 面白うございます is archaic and comical.
A full-text search of the fifth edition of Kōjien will return no examples of です used after an adjective, except for the one in the entry on です. It seems that although Kōjien acknowledges that such usage is considered correct, it neither employs nor actively recommends it.
Further reading (in Japanese)
形容詞の否定形に複数の形があるのはなか on 日本語教師の広場
デス・マス体が書きにくいワケ by ‘tobi’
「形容詞＋です」述語の生起要因についての準備的考察 by 前川喜久雄
カリ活用 on Kotobank