X: To accomplish world unity, we need peace.
O: To achieve world unity, we need peace.
X: A balanced diet is accomplished by eating many different kinds of foods.
O: A balanced diet is achieved by eating many different kinds of foods.
When you accomplish something, you manage to do it or complete it, especially something that gives you satisfaction: 'She felt that she could accomplish more through journalism.'
When you achieve something, you manage to do or obtain what you planned to do or obtain, especially after a lot of effort: 'The company plans to achieve these goals within the next five years.'
Note that achieve is also used to mean 'accomplish': 'By the end of the course, you will feel that you've really achieved something.
