As verbs the difference between should and better
is that
should is
Used to form the future tense of the subjunctive mood, usually in the first person.better is to improve.
As nouns the difference between should and better
is that
should is a statement of what should be the case as opposed to what is the case while
better is an entity, usually animate, deemed superior to another; one who has a claim to precedence; a superior.
As an adjective better is
comparative of good.
As an adverb better is
comparative of well POS=adverb.