In general, “had better” is used when there is a greater sense of urgency or importance. It is also used when the speaker is trying to persuade the listener to take action. “Should” is used when the speaker is simply giving advice or making a suggestion.
Surprisingly, there is no entry in the English Grammar Profile for the phrase ‘had better’. In the English Vocabulary Profile, ‘had better‘ with the meaning ‘should’ is listed at A2.
For example:
You had better get out of this room and back downstairs right away.
The sentence above is a strong warning or suggestion that the listener should leave the room immediately. The speaker is implying that there is danger or something wrong in the room, and that the listener should not stay there.
A search for collocates in COCA of:
had better_RRR
1 START 82
2 PREPARED 59
You had better be prepared to push yourself harder than ever.
Chicago Sun-Times
3 WATCH 52
4 READY 47
5 CAREFUL 32
6 LEARN 31
7 DECIDED 30
I decided I had better get out of there.
Saturday Evening Post
8 HURRY 28
9 WAKE 28
10 SKILLS 19
11 LISTENING 16
12 TASTE 15
13 PREPARE 14
14 PRAY 13
15 RETHINK 6