The main difference is that where bitmap (pixel) images are resolution dependent, vector images are resolution independent. This is basically whether or not the image is "independent from the pixel grid." When you scale up a bitmap image it becomes rough-edged and blurry, but if you scale a vector image (which is based of of mathematical equations) it is the same resolution.
Vector images are common when designing corporate logos because the designer can scale the logo as big as he wants or as small as he wants without changing the initial resolution. This allows to get an image the size of a sticker to the size of a road side billboard.
Main Link
Link used for clarification
Definition of resolution independence
Sorry for using wikipedia, but this was the only link I could find that told me what resolution independence was.
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