남자와 여자 웃습니다 = The man and the woman laugh. => Two subjects both doing the same action, plural verb.
남자가 여자와 같이 웃습니다 = The man laughs with (와...같이) the woman. => One grammatical subject (the man, which is marked by 가), singular verb. The woman is just an accessory. (For want of a better word, sorry.)
it depends on the word order mostly.
- 남자가 여자와 같이 웃습니다 means the man is the sole subject of the verb (as it is marked by the subject particle) and he's doing so with the woman.
- 남자와 여자가 같이 웃습니다 means the man and the woman are both subjects of the verb and they are doing so together.
In the sentence, the man is our topic or someone that we should be focusing on. If we switch them, the listener may think that the woman is our main focus. It's kinda like subject-verb agreement in English. So in English, we would say "The man, together with the woman..." That's the reason why it is "남자가 여자와..."
"남자가" marks the man as the sole subject of the sentence, but in your English sentence the subject is "the man and the girl".
Also, it's weird to translate 남자 as man and 여자 as girl. While 여자 can be girl, there's nothing indicating that the "여자" is younger than the "남자" in the sentence.
In my opinion, a more accurate translation is "The man laughs together with the woman."