What’s the difference?

English distinguishes auxiliaries and modal verbs. Auxiliaries (to be, to have, to do) combine with present participle or past participles or infinitive to form the verb forms of ordinary verbs. Modal verbs combine with infinitives to express ability, probability, necessity, permission, obligation, deduction etc.

auxiliariesmodal verbs
am-is-are 
was-werecan, could, to be able to
has-have-hadwill, won’t, shall, shan’t
to have towill,would, shall, should
short yes/no answermodal mix: (can, could, may, might, must, had to, ought, shall, should, will, would)
tag-questionsmodalities: ability, permission, probability, deduction, necessity, request and obligation