I wrote this post as an answer to question on bunpro forums, but since it has grown in size I wanted more people to see it and learn something useful 
Long time ago, in Japanese nominals (nouns) were used as adjectives, to describe other nouns. They were simply put before modified noun.Eventually Japanese started adding し to those nouns, creating first adjectives. Some of those also gave birth to some static verbs, that’s why we have some static verbs and adjectives that sound similar.
Example: 大人 - “adult” => 大人し - “mature”
The form ending with し was 終止形(predicative form), the form only used at the end of the sentence (and not in all circumstances).
The first adjectives were the ones that represented observable physical condition.For example: 清し (‘clean’, ‘clear’).This group were called く adjectives (due to their 連用形 form)
The other group which also appeared about the same time but was less numerous (originally, since eventually, they have grown in number) also ended with し in their 終止形. They were more subjective, and described feelings and so on.For example 美し (‘lovely’)This groups were called しく adjectives (due to their 連用形 form)
While both ended with し, the major difference was in conjugation:
| Conugation | 清し (く adjective) | 美し(しく adjective) |
|---|---|---|
| 未然形（imperfective form (things that not happened), nai stem) | - | - |
| 連用形(conjunctive form (connects), masu stem) | 清く | 美しく |
| 終止形(predicative form, ending form) | 清し | 美し |
| 連体形(attributive form, noun modifying form) | 清き | 美しき |
| 已然形(realis form (thing that happened, used with ば and ども as in けれども) | 清けれ | 美しけれ |
| 命令形(imperative form) used for orders | - | - |
As you can see, the major difference was that in the case of しく adjectives, し stayed in all conjugations, while in く adjectives case it was replaced.Examples: なし(ない)、べし(nowadays べきだ)、ごとし,
To those conjugations various auxiliary verbs were added. Those auxiliaries were for example む、なる and some other.
You can also notice, that both kinds of adjectives did not have an imperative form or imperfective form. For that purpose verb, あり (nowadays ある) had been added to 連用形 which gave くあり ending. This was shortened to かり and conjugated for the missing forms. This gave birth to modern conjugations like past form of adjectives for example:
| Conugation | 清し (く adjective) | 美し(しく adjective) |
|---|---|---|
| 未然形（imperfective form (things that not happened), nai stem) | 清から | 美しから |
| 連用形(conjunctive form (connects), masu stem) | 清かり | 美しかり |
| 終止形(predicative form, ending form) | - | - |
| 連体形(attributive form, noun modifying form) | 清かる | 美しかる |
| 已然形(realis form (thing that happened, used with ば and ども as in けれども) | - | - |
| 命令形(imperative form) used for orders | 清かれ | 美しかれ |
In other words, it comes from combination of く form + ある (to exist).
くあった → かった
Eventually, い-sound change happened in attributive form, and the k in き was dropped. (Similar change happened to some verbs like ござります->ございます)
So:
| Conugation | 清し (く adjective) | 美し(しく adjective) |
|---|---|---|
| 連体形 old | 清き | 美しき |
| 連体形 new | 清い | 美しい |
And this is from where we have the modern forms, eventually, the 終止形 (the predicative form) stopped being used, and the 連体形 (attributive form) took its role which made it possible to use the same form to when both modifying nouns and at the end of the sentence.
This is why some adjectives ending with しい describe feelings more often, and the ones just ending with い describe some observable phenomenonsOf course, some adjectives retained some of their original conjugations, like ごとし . You can also see sometimes なし (old predicative form) なき (old attributive form) being used instead of ない. が follows なき because attributive form could be used to nominalize in classical Japanese, just like modern の or こと.
Example:サービス業のマネージャーを任されている私にとって、休日などはあってなきが如し。
For me, who has been entrusted as the manager of the service sector, something like days off are virtually nonexistent.
