A quest is a journey towards a goal used in mythology and literature as a plot. Quests can be found in the folklore of every nation. In literature, the objects of quests require great exertion on the part of the hero, and the overcoming of many obstacles, typically including much travel.

This travel also allows the storyteller to showcase exotic locations and cultures, which may, indeed, be the writer's objective if not the characters.

taken from Wikipedia

Summary

In Aion, quests consist of simple tasks that are not necessarily related to the overarching plot of the Aion Universe. For instance, an NPC might ask you to collect 15 pelts from a certain mob in order to make a special leather item. But this has no bearing on the major conflicts of the story.

Missions are major quests that advance the plot of the game.