Garden box vs just growing in the ground

Why not just dig out an area, mix the existing soil with compost or manure, and put some mulch on the top?

Why not? It would work in many cases.

Some reasons to make a box raised bed

- ergonomics (better working height)
- protection from critters
- visual appeal

any others?

For part of my property, I chose raised garden beds due to having a bunch of junk located in the ground of the area. Large rocks, gravel, miscellaneous metal and glass from when a stable existed on site. By having a raised bed, I have better control over the soil characteristics.  

Good point Timothy - you can even put a raised bed on concrete if that is all you have in your yard.

I know a lot of people garden in raised beds because of the perception that you can't improve the clay soil around here. And it is very heavy, but it can definitely can be improved by mixing manure or compost into it.  Though I have to admit that that doing so is a lot of work because clay is no fun to dig in and turn over.  It almost seems easier to just build on top of it.

Edit:  I should add that I'm well aware clay can be an amazing growing medium but it just takes SO MUCH organic matter to get it to that point.

I should also add that I keep adding very expensive raised beds because I am far more likely to use them efficiently than if I just  plant in the ground. I think it's almost a mental thing. The define lines make it obvious that I should be using every square foot and not wasting the money that the beds cost.