화(가) 나다 means "to be angry"
화(를) 내다 means "to express anger to someone"
그는 그녀에게 화가 났다 means "He was angry because of her" - It is uncertain whether she noticed his anger or not.
그는 그녀에게 화를 냈다 means "He was angry and expressed anger to her" - It is certain that she noticed his anger.
The following is possible.
그는 그녀에게 화가 났지만 그녀에게 화를 내지 않았다.
He was angry because of her, but din't express anger to her.