最后：It means "at last" "in the end" "finally" When we use this word,it usually shows the reslut for one thing.
eg:I went shopping yesterday,finally I don't buy anything
终于：It is similar to 最后。But this word used to express great emotion or desire for something you want or you are doing.
Finally I buy tht ice-cream(great desire for ice-cream)
Finally I finished homework(It show emotion I feel very hard,maybe there is too much homework;maybe I want to play football with my friends,so I finish my homework in a hurry at first)