Immortal, deathless, undying, unfading mean not subject to death or decay and, hence, everlasting. With the exception of immortal, all of these words are chiefly in poetic use and are distinguishable especially in their connotations and applications.
Basically immortal implies little more than exemption from liability to death and is usually applied to the soul or spirit of man.
Sometimes immortal equals eternal, but more frequently it keeps close to the basic sense in being applied to something comparable to the soul in that it lives on in fullness of vigor after its maker or possessor has died.
Deathless also implies incapacity for death; it is seldom applied to the soul but rather to immaterial things that transcend the limitations of mortal existence.
Undying is applied chiefly to emotions or passions marked by such intensity or vitality as to be or to seem incapable of extinction while life lasts.
Unfading often comes close to undying in meaning but connotes persistence of brightness or bloom rather than of intensity.