First off, the difference between the two is quite simple. Pixelated files are made of thousands of "pixels", and when these images are zoomed in on, or blown up and printed, they tend to appear very bitty (and look like a whole bunch of squares). A common pixelated file format would be a jpeg (which is usually the format your digital camera would take pictures in). A vectored file is made up of mathematical paths and shapes, and therefore can be scaled as large, or as small as needed. A common vectored file format would be an eps (encapsulated post script) file, which a logo, or solid graphic elements are usually designed in.
One benefit of vectors is file size. They tend to be quite a bit smaller than a rasterized file of similar nature...especially when dealing with high resolution print ready files. Another benefit of the vector file format is colour seperation. It is easier to prepare a vector file for printing, when the job involves specific corporate PMS (Pantone) colours over and above the basic CMYK (Cyan, Magenta, Yellow and Black). It is for these reasons that a corporate logo is almost always designed in a vectored format - and to ensure that it will have consistent reproduction at any scale.