If an area is messy, it means it is disorganized, with many various objects all over the place. A messy area needs to be organized and things put in their proper places.
My desk is so messy - there are piles of documents everywhere. I can't find anything I need.
But if an area is dirty, it has accumulated dirt/dust and needs to be cleaned or washed. After you eat dinner, the plates, forks, knives, and spoons are dirty. After you exercise and sweat, your clothes are dirty.
When my computer gets dirty, I clean it with a special solution that won't damage the screen.
We drove down a very muddy road and now the car is all dirty.