추운 and 추워 both come from the word 춥다. 춥다 is the dictionary form of 추운 and 추워. 춥다 means "to be cold". 추운 means "cold" (adjective). 추워 means "is/are cold" (descriptive verb). 추운 is in the noun-modifying form. That means it's an adjective and a noun HAS to come after it. 추워 is just in a sentence form and comes at the end of a sentence. A noun would come before it. You could also add 요 to make it polite. 추운 물 - cold water -> just a little word phrase. you would need to add more after it in order to make a complete sentence. -> 추운 물이 맛이 있어요. - The cold water tastes good. 물이 추워요. - The water is cold. -> complete sentence. good. This pattern applies to all adjectives. 뚱뚱하다 - to be fat (dictionary form) 뚱뚱한 - fat (adjective) 뚱뚱해(요) - is/are fat (descriptive verb) 뚱뚱한 고양이 - fat cat -> INCOMPLETE 뚱뚱한 고양이는 상추를 안 먹어요. - Fat cats don't eat lettuce. -> COMPLETE SENTENCE 고양이가 뚱뚱해요. - The cat is fat. -> COMPLETE SENTENCE

추운 (x) - cold SOMETHING; (x) 추워 - SOMETHING IS cold.