구름이 끼다 = clouds came in; the sky is cloudy. 흐리다 = It is overcast/cloudy = There is no visible sun. So you say 구름이 끼다 if you can see the sky and lots of clouds forming. 흐리다 describes an overcast day when the sky is just gray overall so that you can't even see any clouds. 흐리다 is the more common expression that goes against 맑다 (clear) / 화창하다 (sunny). 구름이 끼다 usually refers to the change, like the sky was clear but then it became cloudy.

Can't say how thankful l am