There is a difference in tone and level. "Errand" is an everyday word. I run errands all the time. I need to pick up some things at the supermarket today; that's an errand. I need to return some books at the library today; that's an errand.
A "mission" involves a major and important goal. Except as a joke, I would never say "My wife has sent me on a mission to find caraway sea salt Wasa thin crisps."
It often suggests a military goal, particularly one where the goal is specified, but not the particular directions for achieving it.
"Mission" is used for organizations, in a piece of jargon which I happen to dislike, the "mission statement." For example, "Our mission is to provide children from birth through age 12, living in homeless or low-income situations, with the essential items they need to thrive – at home, at school and at play."
The oldest meaning of "mission" is the one found in the word "missionary." Missionaries are people who are sent long distances, to spend years of their life in a distant place; their mission is to get people to accept their faith.