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
vs.
You'd better wear a coat, so you don't get frostbite.
English auxiliary verbs combine into more complex constructions according to wholly inflexible rules: the sequence is always
- the modal component first (if it is present), with the following verb in its infinitive form
- the perfect component next (if it is present), using the auxiliary HAVE, with the following verb in its past participle form
- the progressive component next (if it is present), using the auxiliary BE, with the following verb in its present participle form
- the passive component next (if it is present), using the auxiliary BE, with the following verb in its past participle form
- the lexical verb is always the last.
Each construction is thus marked with a specific auxiliary verb, HAVE or BE, and there is a ‘ripple’ effect: the form (present or past participle or infinitive) of each verb is determined by the preceding component.
Note that the English 'modal' verbs can/could, may/might, must, shall/should, will/would are defective: they occur only in finite forms and have no non-finite forms (infinitives and participles).
Since a modal verb cannot be cast in the appropriate non-finite form, it cannot act as complement of a prior modal verb or auxiliary: except when two modals are conjoined (You can and should tell him), you can have only one modal verb in a clause, and it must be the finite first verb.
You occasionally hear paired modals in dialect speech, but this is emphatically non-standard and is generally taken to mark the speaker as uneducated.
The work-around for situations which call for "dual modalities" is to employ a periphrasis for the second modal:
*You might could do it → okYou might be able to do it.
A periphrasis like be able to VERB or need to VERB or be obliged to VERB in effect "restarts" the verb chain: a complex construction with its first element cast in the 'infinitive' may follow the to:
If you had planned better you might have been able to have been working offsite when he arrived.
In this case, however, the first element in the new chain cannot be a modal, because modals cannot be cast in the infinitive.
Best Answer
"Can" is referring to an actual set of circumstances in the present (or future), or timeless:
"Could" is either referring to the same in past time:
or to a hypothetical at any time: