Quest and pursuit are two English words that have almost the same meaning. The truth about these words is that they are synonymous. This means, there is no much difference between quest and pursuit. However, there are still little differences between them. Quest is an effort or a journey in pursuit of a goal. Quest, in its actual meaning, comes with a powerful force in someone who is ready and dedicated at achieving an already set goal. Pursuit is the act of pursuing when you pursue something, it means you are following that thing with the intent to chase, capture or harm it.
Although, the pursuit is also used to show the readiness of someone to go after his dreams and goals, it has more negative connotations than the positive ones. Quest, on the other hand, is just like a strong desire for something you genuinely want. Quest can be used as a noun, it can also be used as a verb, but the pursuit is mostly used as a noun.