Japanese verbs have the same form whether predicative or attributive. (The only exception is the copula, which is da or desu when used predicatively and na when used attributively.)
However, this is not the case in the Korean language. Korean puts an attributive verb ending(관형사형 전성어미) to make a verb or an adjective behave as an attributive, and these endings have innate tense that is different from the verb tense. For example, 죽다(to die) in the attributive past tense form is 죽은, while in the verb past tense form is 죽었-.
그가 어제 죽었다. "He died yesterday."
어제 죽은 그 "he who died yesterday"
Korean adjectives are considered present tense by default. This explains why adjectives do not take present-tense verb endings in the predicative position.
산이 높다. "The mountain is high."
*산이 높는다. (X)
This also applies to the attributive form. It does not take present-tense attributive verb ending "는". Instead, it takes ㄴ/은 for the present form, which is a past form for verbs.
*높는 산 (X)
높은 산 "a high mountain"
Attributive adjectives take -던 for the past-tense, which can be taken by a attributive verb too, though the meaning is a little different from 은.
높던 산 "a mountain that was high"
죽던 그 "he who was dying"
죽은 그 "he who is dead"
Let's go back to the Middle Ages. Middle Korean had no tense, like Chinese. The now extremely common verb tense pre-endings -었-, -는-, -겠- were nowhere to be found. Instead, Middle Korean used moods. There were four major moods in Middle Korean, and each mood mapped to several tenses.
As you can see, the attributive form is just -ㄴ added after the predicative endings. (-린 is special, the ㅣㄴ dropped very often, and used like -ㄹ most of the time.)
The indicative mood(직설법) was a grammatical device which was used to objectively describe something that is happening, almost always in the present. It normally couldn't be put on an adjective, because adjectives describe a state, and a state can't be actively happening. But in the case when it did, the adjective was used like a verb, meaning "to become such state".
The aorist mood(부정법), sometimes called infinitive mood, primarily indicated a fact, which happened in the past when the word is a verb, and is in the state right now when it's a adjective.
The retrospective mood(회상법) was used to describe a recalled memory, which can be both past and present.
Presumptive mood(추측법) was used to describe a supposed situation or a will, which has little to do with the truth. So it was often used to describe the future.
Then starting from the 18th century, Korean shifted to a new set of TAM(tense-aspect-mood) system. Past -었-, present -는-, and future -겠- was introduced, and the original mood system got largely abandoned.
But not completely. In the attributive position, the mood system partially survived. But the current tense system overpowers the historical mood system, so we say that -은/는 (directly derived from Middle Korean aorist -ㄴ) is a past tense on a verb and present tense on an adjective, when really, it's just a remnant of the original aorist mood.
So that's why adjectives can't take 는 (derived from Middle Korean indicative -ᄂᆞᆫ-) in Modern Korean(except for 없는). According to the Middle Korean grammar, *높는 산 would mean "a mountain which is becoming higher", which is clearly not what we want to describe. Also, that part of the grammar is no longer in use in Modern Korean(it's wrong), and is replaced with something like 높아지는 산, with -아/어지- which makes an adjective into a verb.