Differences between "had better and should"

Had better is more stronger than should.
It's very cold here.We had better go inside.
"We should go inside" is more of a suggestion, and "You should go inside" is more of a demand. 
You had better take your pills. (I insist that you take your pills.)
You should take your pills. (It would be a good idea for you to take your pills.)
We use “had better” to give advice about specific situations, not general ones. If you want to talk about general situations, you must use “should”. 
When we give advice about specific situations, it is also possible to use “should”. 
However, when we use “had better” there is a suggestion that if the advice is not followed, that something bad will happen. 
"Had better" is used by a person in a superior (or at least equal) position who is giving advice to someone whom he perceives as having no real option but to follow his advice.