Connotation is very different from symbolism, yet, connotation may at times be used in constructing symbolism. Both connotation and symbolism are identified as components of diction. Connotation would relate most closely to the vocabulary component of diction, while symbolism would related most closely to the abstraction component (concrete versus abstract adjectives, verbs, images).
Connotation is defined as the culturally based secondary meaning of a word that gives it meaning beyond its literal definition. An apt and often used example is the difference between the denotations (i.e., definitions) and connotations of house and home. You may own a dozen houses, yet you may have only one home, one place accumulating your personal experiences and possessions. The connotation, then, of house is a place where someone may dwell . The connotation of home, then, is a place where you live your life and have fond connection and relationships.
Symbolism
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Symbolism is a different sort of thing altogether. Symbolism takes a word, idea, or object and uses it to create a new meaning for the purpose of adding depth to a theme, image, action, problem, or character. One particular difference between connotation and symbolism is that connotation is culturally based (e.g., perhaps in Borneo the idea of house cannot be contrasted to that of home) while symbolism is invented by the author (though there may be some conventional symbols that are well used and well known, like a rose for love).
As an example, I'll create a symbol for this answer I'm writing: The dicot leaves watched over the writing of the answer. Now I'll speak of it with a connotative word: The minor treatise was posted. The symbol of dicot leaves emphasizes the two parts of the answer. The connotation of treatise suggests a detailed answer.
Applying this to Crime and Punishment, the opening paragraphs have a very good example of a connotative vocabulary choice. We are told Raskolnikov lives in a "garret" in a "five-storied house":
His garret was under the roof of a high, five-storied house and was more like a cupboard than a room.
"Garret" has a very distinct connotation. Five-storied houses were understood to be microcosms of society. The ground floor housed wealthy people who had carriages and entertained friends. The first floor above ground level housed professional people who were financially comfortable and well respected. The upper floors housed workers, like clerks, then artisans, then manual laborers. The attic garrets housed people too poor to afford anything else. Thus the connotation of "garret" is a dwelling for very, very poor people. The poverty connoted by garret is confirmed by the "five-storied house."
A symbol occurs later on when Raskolnikov's early life is being described. The horse being beaten when young Raskolnikov is walking with his father symbolizes several things, including the beating Raskolnikov receives in life because of his unselfish giving to others who are in need.
"Father, father," he cried, "father, what are they doing? Father, they are beating the poor horse!"
Compare the example of connotation with the example of symbolism and you'll readily see the difference between them: horse has no underlying connotative meaning yet it can symbolize elements of life, while garret has a decided connotation related to destitution.