なければならない・べき: the difference
なければならないN4
Meaning
Must do, Have to do
Past papers
Tested 2 times in past papers
Example
先生の話を聞かなければならないよ。
You have to listen to the teacher.
べきN3
Meaning
Should, Ought to
Past papers
Tested once in past papers
Example
税金を払うべきです。
You should pay taxes.
How to tell them apart
「～なければならない」 is for rules, laws or real necessity: 「パスポートを更新しなければならない」.
「～べきだ」 is the speaker's strong judgment of what is right or the best thing to do. It is stronger than 「～ほうがいい」, which is just friendly advice.
Don't tell a boss or customer 「～すべきです」; it sounds arrogant.