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Freud

Freud believed that the principal driving force behind men and women’s activities (libido) was repressed sexuality which led to pathological conditions. Jung believed that sex constituted only one of the many things that drive humans. More importantly, humans are driven by their need to achieve individuation, wholeness or full knowledge of the self. Many emotions drive humans to act in psychologically unhealthy ways, but all these ways were a longing for the desire to feel complete. Unlike Sigmund Freud, Jung thought spiritual experience was essential to our well-being. The theoretical divergence between Jung and Freud on the nature of the libido led to a break in the friendship between the two men, both stating that the other was unable to admit he could possibly be wrong.

Freud believed that in dream analysis, the manifest content (i.e., what we would call a description of the dream as we remember it; the images and dialogue) should be studied to uncover the latent content (what the dream is "really" about). He believed that the symbols were personal, perhaps things a person saw and connected to an idea. Jung believed that dream symbols are universal. Thus, it is from his ideas that dream symbol books were created.

Freud spoke of the id (the will to pleasure and gratification in each of us), the super-ego (the rule based, "conscience") and the ego (the "self" as we understand the term -- a middle man for the will of the id and superego). Jung believed in arch-types. For example, his version of the id is the "shadow." Like Freud, he believed that conflicts arise because of these competing wills.

Examples on Freud's the Sexual Interpretation of Symbols

1. Freud provides a psychoanalytical interpretation of da Vinci's The Virgin and Child with St. Anne. According to Freud, the Virgin's garment reveals a vulture when viewed sideways. Freud claimed that this was a manifestation of a "passive homosexual" childhood fantasy that Leonardo wrote about in the Codex Atlanticus, in which he recounts being attacked as an infant in his crib by the tail of a vulture.Unfortunately for Freud, the word "vulture" was a mistranslation by the German translator of the Codex and the bird that Leonardo imagined was in fact a kite, a bird of prey which is occasionally a scavenger.
2. Ida a patient of his recounted two dreams.one of them was: a house was on fire. My father was standing beside my bed and woke me up. I dressed quickly. Mother wanted to stop and save her jewel-case; but Father said: 'I refuse to let myself and my two children be burnt for the sake of your jewel-case.' We hurried downstairs, and as soon as I was outside I woke up.Freud reads this dream as referring to Ida Bauer's sexual life - the jewel case that was in danger being a symbol of the virginity which her father was failing to protect from Herr K. 
3. Every clan has a totem (usually an animal, sometimes a plant or force of nature) and people aren't allowed to marry those with the same totem as themselves. Freud examines this practice as preventing against incest. 

Jung

Jung added to Freud's definition of the unconscious that each individual also possessed a collective unconscious, a group of shared images and archetypes common to all humans.

In Jung's psychological framework, archetypes are universal patterns and images that derive from the collective unconscious, Being unconscious, the existence of archetypes can only be deduced indirectly by examining behavior, images, art, myths, religions, or dreams and being part of the collective unconscious means it is shared among all people and all cultures.Jung described archetypal events: birth, death, separation from parents, initiation, marriage, the union of opposites; archetypal figures: great mother, father, child, devil, god, wise old man, wise old woman, the trickster, hero, properties shared among humans in which a person becomes a certain archtype for a given part of his life.

Jung believed that this journey of transformation, which he called individuation, is at the mystical heart of all religions. It is a journey to meet the self and at the same time to meet the Divine. Jung considered individuation, a psychological process of integrating the opposites including the conscious with the unconscious while still maintaining their relative autonomy, necessary for a person to become whole.

Archetypes seek actualization within the context of an individual's environment and determine the degree of individuation. Jung also used the terms "evocation" and "constellation" to explain the process of actualization. Thus for example, the mother archetype is actualized in the mind of the child by the evoking of innate anticipations of the maternal archetype when the child is in the proximity of a maternal figure who corresponds closely enough to its archetypal template.

Psychological Interpretation of Myth

Because the unconscious “speaks” in another kind of “language” than the conscious, myth serves as an intermediary: a means for the unconscious to communicate its symbolic meaning.The whole of mythology could be seen as a sort of projection out of the collective unconscious. Jung then asserts that the similarities among myths are the result of the projection of the collective rather than the personal unconscious onto the external world. Finally, he comes to the conclusion that myth originates and functions to satisfy the psychological need for contact with the unconscious--not merely to announce the existence of the unconscious, but to let us experience it.

Psychological Interpretation of Dreams

Jung takes dreams as the analogue to myths; both myths and dreams arise from the collective unconscious, connect unconscious to conscious and encourage us to pay attention to what arises. Jung considers dreams a more pristine manifestation of the unconscious (coming from the personal unconscious, not ordered and generally unintelligible and irrational) and therefore their interpretation requires less reconstruction than the interpretation of myths. Only archetypal dreams, identified by their mythical content, originate from the collective unconscious as myths do. Myths however are closer to the unconscious than dreams.

Psychological Interpretation of Gnostic

Jung understands the emergence of the Demiurge out of the original, unified monadic source of the spiritual universe by gradual stages to be analogous to (and a symbolic depiction of) the emergence of the ego from the unconscious

Psychological Interpretation of Religion

For Jung all myths are psychological; many religious myths, for instance the Christian myth about Jesus, are about man’s life-long quest for individuation.

Psychological Interpretation of Alchemy

Jung asserted that the medieval alchemists were unaware of the natural process of psychological transformation which went on in their subconscious.  Therefore, they projected this process into their experiments as a science of the soul.  In other words, they projected an inner process outside of themselves.

In psychology the four-fold nature of the prima materia is expressed by the four functions which correspond with the alchemical elements: intuition (fire), thinking (air), feeling (water), and sensation (earth). In Jungian theory we have a dominant function and limited access to one or two others, but the fourth function is inaccessible, elusive, maladapted, or hard to integrate.  It is what keeps us from "getting it all together."  Thus, we are out-of-balance.Balancing the four functions means achieving an integrated personality, harmony and well-being.

The four major stages of the alchemical magnum opus, creating the Philosopher's Stone can be Interpreted physiologically also in which these four alchemical steps are be taken as analogous to the process of attaining individuation

1. Nigredo, or blackness, elements had to be cleansed and cooked extensively to a uniform black matter. In analytical psychology, the term became a metaphor 'for the dark night of the soul, when an individual confronts the shadow within'
2. Albedo or whiteness, the alchemist undertakes a purification by washing away impurities. In analytical psychology, the term became a metaphor for the unconscious contrasexual soul images; the anima in men and animus in women. It is a phase where insight into shadow projections are realized, and inflated ego and unneeded conceptualizations are removed from the psyche.
3. Citrinitas or yellowness. In analytical philosophy, citrinitas stood for the dawning of the "solar light" inherent in one's being, and that the reflective "lunar or soul light" was no longer necessary becoming the wise old man (or woman) archetype.
4. Rubedo or redness" signalling alchemical success, the merging of ego and Self , it is the Self archetype which has achieved wholeness.

Rosarium Philosophorum

In a study Jung employed the ten pictures illustrating the opus of alchemical transormation contained in a classic called Rosarium Philosophorum (Rosary of the Philosophers), where the dual powers of the "King" and "Queen" are shown to undergo a number of phases of their own mystico-erotic relationship and eventually unite in a new, androgynous being, called in the text "the noble Empress". The series of images begins with that of the mercurial fountain, symbolizing the aroused energy of transformation and continues with the meeting of the King and Queen, first fully clad and later having relinquished their garments. The lovers thus confront each other with their personae and defenses, but proceed to a meeting in "naked truth". The partners then immerse themselves in the alchemical bath, thus allowing the force of love to engulf their conscious egos, blotting out rational and mundane considerations. While in this state of passionate engulfment the psychosexual union (coniunctio) takes place. But, contrary expectations, this union, which initially brought forth a newly formed androgynous being, results in death. The spiritual result of love is not viable and, having expired, undergoes decomposition. It is at this point that the force of commitment to the process (though not necessarily to a particular partner) becomes all-important. By not abandoning the transformational work, the soul of the dead androgyne ascends to heaven, i.e., to a higher level of consciousness, while the body is washed in celestial dew. Soon the departed soul returns to its earthly body, and the reanimated corpse stands in its full, numinous glory for all to see. A new being is born which is the promised fruit of love, the transformed consciousness of the lovers, formed of the opposites, which are now welded into an inseparable imperishable wholeness. The alchemy of love has reached its true and triumphant culmination.

In The Psychology of the Transference, Jung has shared with the world his uniquely practical insight not only into the psychological mechanism of love but into the process of the reconciliation of all opposites - emotive, intellectual, physical, and metaphysical. At the root of all being, so he intimates, there is a state wherein physicality and spirituality meet in a transgressive union. Synchronistic phenomena The tensional relationship of the opposites remains the great operational mechanism of manifest life and of transformation, this relationship exists within the context of a unitary world-model wherein matter and spirit, King and Queen, appear as aspects of a psychoid realm of reality.

Unicorn Example

References to the cup made from the horn of the unicorn in Greek, Chinese, and Christian traditions are
presented. It is noted that the unicorn cup is related in some way to both the Eucharistic Chalice and the
vessel used in divination. The secret of the cup is also the secret of the unicorn horn and stands for the
essence of the unicorn as bringer of health, strength, and life. The dual nature of the horn is analyzed. As
a symbol of vigor and strength it has a masculine character; as a cup it is also a feminine symbol. Thus
the cup becomes a " uniting symbol" expressing the bipolarity of the archetype