Notice that Paul separates "immortality" from "eternal life" as though they are different. The words assuredly share a common idea, that is, both indicate a long, enduring period. Immortality simply means "unending existence" because the being does not corrupt, decay, and die.
However, "eternal life," as used by the Bible's writers, includes something "immortality" does not, introducing a shade of difference between the two words. Unfortunately, in many minds, "immortality" corresponds exactly with "eternal life." They are not the same.
Perhaps a good way to illustrate this is to refer to the Greek myths with their pantheon of gods. In these myths, the gods had immortality but—by biblical definition—not eternal life. This is because immortality speaks only of endless life, not its quality. The Greek gods acted, reacted, and had passions and attitudes just like human beings, mere mortals, whereas eternal life in the biblical sense is life lived the way the true God lives it. It indicates the totality of life, which, as we will see, we already possess in principle. To put it into a more human setting, eternal life is to live life endlessly according to the will of God.
John 5:24 helps us to begin to understand when Jesus says, "Most assuredly, I say to you, he who hears My word and believes in Him who sent Me has everlasting life, and shall not come into judgment, but has passed from death into life." Notice that he who believes has already passed from death to everlasting life. We can connect this to Ephesians 2:1: "And you He made alive who were dead in trespasses and sins." Before repentance and conversion, God views us as dead even though we are physically alive.
Though we possess animal life, before God's calling we are totally unaware of the spiritual life of God, even as those who are physically dead are unaware of the pleasures, cares, and amusements of the living. They hear no music, enjoy no food, can see neither beauty nor ugliness—they are unaware even of people trampling on their graves! Before conversion, we are likewise unaware of the spiritual life of God, the beauty of holiness, and the joy, power, abundance, peace, honor, and glory of that life. Conversion is a life slowly expanding into a new dimension that we never knew existed before—everlasting or eternal life.