¿Cuál es la diferencia entre refer y consult y appeal y apply ?Puedes dar oraciones como ejemplo.

Refer means to look into something for information. Eg. For more examples, refer to your textbooks.
In legal terms, 'refer' is used to hand over or submit information for consideration. Eg. refer to the panel.

'to consult' is to discuss with and ask for advice. eg. Consult a doctor.

An appeal is a request for support. In law, an appeal is when someone approaches a higher authority to question the validity of the ruling by a preceeding authority.
(appeal could also mean attractiveness)
to apply is to request formally.
(other meanings: to put it over a surface or to put it into action)