No. People of different eye colors see the world exactly the same as long as they don't have eye defects. Their eyesight can't be affected by the colors of their eyes. The difference of eye colors is caused by the difference of amount of melanin in iris as a result of evolution and natural selection.
Not all can affect vision. Eye colors are determined by genes, ages and growing environment. If someone are not color-blindness, they see the world almost the same, but a lightly difference still exists. Someone ever told me that people who have blue eyes will see color much brighter than dark eyes.
Actually different people with different eye color will see the things in the world similar, with no big difference. The color of the eyes is determined by the genetic genes which will not change easily. What you see the world is related with your vision. In order to get the beautiful color of the eyes, a lot of people now wear the colored contact lenses which will be workable.