區別就是兩者詞性是不一樣,具體的不同如下
over表結束了是形容詞
finish是動詞,但be over=finish,中文意思是結束,完成了。
例如
Our budding romance was over.我們剛剛萌發的浪漫愛情結束了。
=Our budding romance finished 
我們到達時,會議已經結束了。By the time we arrived the meeting was over.
怎麼區別over和finish
over 與finish(結束……)
over作此意講一般是副詞,前面必須有動詞才能做謂語,一般結構是be over. 例如:School is over, I went straight home.
而finish是動詞,後面接名詞或動詞的ing形式。例如:
I finished my homework, I went to bed.
或I finished doing my homework, I went to bed.