总算 means that the speaker feels very happy that the wish comes true after the long waiting.

下了六天的雨，今天总算晴了。The weather is fine at last after the rainy day lasted six days.

考试总算结束了。The exams were over at last. 

终于 The situation appeared after waiting or a series of changes.

经过数十年的奋斗，他终于成就了大业。

面对证据，罪犯终于在法庭上认罪了。

Sometimes, "终于" can instead of "总算"，and means the wish come true finally.

妈妈寄给我的包裹终于到了。 

So "终于" states the fact, and "总算" express the good mood.