最后 and 终于 are very similar in use but they are different! They are different both grammatically and semantically.
他一生中最后的14年是在退休中度过的。He spent the last fourteen years of his life in retirement. (最后 as an adjective)
Finally, the camera’s field of view can be zoomed in and out. 最后，摄像机的视野可以分别放大或者缩小。(最后 as an adverb)
哦，糟糕，在中途中断转播啊，人家想看到最后嘛。Oh, no! They end the broadcast before the game is finished. I wish I could see the end. (最后 as a noun)
bú guò , gé luò fú xiǎo jiě zhōng yú jué dìng hé tā tán tán zhè ge jiǎo de tā xīn shén bù níng de wèn tí 。Finally, however, Miss Glover decided to speak with him on the subject that distressed her. (终于 as an adverb)
“最后” is used for the order of time or space.
这将是我最后一次干这种工作了。 Zhè jiāng shì wǒ zuì hòu yī cì gàn zhè zhǒng gōng zuò le. It is the last time I will take on this sort of work (time).
他坐在最后一排。Tā zuò zài zuì hòu yī pái. He is sitting in the last row (space).
最后，他鼓足勇气走进那间黑屋子。 zuì hòu , tā gǔ zú yǒng qì zǒu jìn nà jiān hēi wū zǐ 。He at last spunked up enough courage and went into the dark room. (time)
“终于” is used in a context in which there are a series of changes, waiting, or expectation before the final happening of a thing.
他终于到家了。Tā zhōng yú dào jiā le. He made his way home at last (waiting or expectation).
警察终于在那些罪犯正准备上船时把他们抓获了。 jǐng chá zhōng yú zài nà xiē zuì fàn zhèng zhǔn bèi shàng chuán shí bǎ tā men zhuā huò le 。The police finally ran down the criminals as they were about to board the ship. (waiting or expectation)