Since both adjectives and verbs all ends with 다, is there any grammatical differences between them. Is it only a difference that we introduced because in English we both have adjectives and verbs and they are very different?
One big grammatical difference between adjective verbs is in the participle endings (which turn the verb / adjectives into modifiers).
For verbs, there are 4 participle endings: Past -ㄴ/은, Present -는, Future -ㄹ/을 and respective past -던:
밥 먹은 사람 (The person who ate the rice)
밥 먹는 사람 (the person who eats rice)
밥 먹을 사람 (the person who will eat rice)
밥 먹던 사람 (the person who was eating rice)
For adjectives, there are only 3 participle endings: Past -던, Present -ㄴ/은, and future -ㄹ:
아주 춥던 그날 (that very cold day)
추운 날 (a cold day)
추울 것 같다 (it seems cold)
That is, the past participle for verbs is the same form as the present participle of adjectives, and adjectives use the retrospective past participle as the past participle.
Another major difference is the semantic meanings of some endings. For example, the 아/어지다 ending differs in meaning depending on whether it is added to a verb or adjective. After an adjective, it means "become -". After a verb, it makes a passive form:
추워지다 (become cold)
만들어지다 (be made); 쓰여지다 (be used)