zuìhòu最后 and zhōngyú both means ‘finally’, but zuìhòu最后 emphasizes the last part of the process while zhōngyú终于 means the consequence appears after a long process, and the consequence is frequently what the speaker expect. For example,
1)首先, 其次,最后
shǒuxiān,qícì,zuìhòu
First Then At last
2)我终于成功了
Wǒ zhōngyú chénggōngle
I finally succeed
‘I finally succeed.’