The primary difference between the two is in the implication of the result.
"You should" carries the connotation that if you do as suggested, the outcome will be favorable; it puts the focus of the statement on a positive consequence.
You could say that it is an encouragement to engage in good behavior.
"You'd better", on the other hand, carries the connotation that if you fail to do as suggested, the outcome will be unfavorable; it puts the focus on the negative consequence. You could say that it is a warning about engaging in bad behavior.
If the phrases were extended, you would almost always see something along the lines of
You should wear a coat, so you can stay warm
You'd better wear a coat, so you don't get frostbite.