Everything means "all things", whereas anything means "any one thing".
Example 1: You can order anything you like. (Meaning any one object)
Example 2: I'm starving! I could eat everything on the menu. (Meaning all the food on the menu, not just one dish)
Interestingly, the word "anything" is also often used in negatives and questions. You can find some interesting examples here:
I hoped this helps?