What is the difference between "mission trips" and "service hours"

A Sunday school class asked that I come and share with them what the youth do for missions and how that class could help and support. The conversation began with a foundational discussion as to why do mission work at all. Matthew 25's Parable of the sheep and goats was the text that rooted the discussion.

Eventually the question was asked to the class, "Why should Christians do mission work?"

"To help people."
"It is a good thing."
"Make the world better."
"It is good for my soul."

These were some of the responses that were given. The next question was asked, "What is the difference between these answers and answers given by an atheist? Surely an atheist could say the same things we have already stated and they do not affirm a deity. So what makes the work of the atheist and the work of the Christian any different?"

The only response came in the form of a question, "Because Jesus tells us to?"

Again, even an atheist has a model for their behavior. So the question remains, what makes our work in the world "Christian"?

I am not sure of the answer, but I would submit in my understanding the work of the Christian in any service/mission/volunteer is to help usher in the Kingdom of God (KoG). I am assuming that the KoG looks different from a world that an atheist would look to help create. Not that they are in competition or at odds with each other, but they are not the same. Perhaps they are complementary, like colors on a color wheel, but they are not the same.

So I guess, what I am proposing is the work of the Christian is not rooted in helping people but restoring creation. Not because it is good, but because it is very good. Not to make the world a better place, but making the world God's place. And not because it is good for our soul, but because it is the reason for our soul.