From an grammatical viewpoint, the difference between "춥다" and "추워하다" is that "춥다" is an adjective and "추워하다" is a verb, formed with "춥다 + 하다".
좋다, 좋다+하다, 좋아하다.
덥다, 덥다+하다, 더워하다
춥다, 춥다+하다, 추워하다
놀랍다, 놀랍다+하다, 놀라워하다.
Otherwise, on a reading viewpoint,
그 여자아이가 추워요. (It is true) the girl feels cold.
그 여자아이가 추워해요. (I think/believe) the girl feels cold.