In obsolete terms the difference between promise and deal
is that
promise is bestowal or fulfillment of what is promised while
deal is a division, a portion, a share.
In transitive terms the difference between promise and deal
is that
promise is to commit to something or action; to make an oath; make a vow while
deal is to sell, especially to sell illicit drugs.
In intransitive terms the difference between promise and deal
is that
promise is to give grounds for expectation, especially of something good while
deal is to handle, to manage, to cope.
As an adjective deal is
made of deal.