As seen in the example above, logos, because they are used so often in so many different environments, should always be vector format. Any art that needs to be easily resized should be vector.
Vector art is suitable for any image that uses flat swaths of color. Maps, icons, cartoon-style drawings — they’re all usually in vector format. If the Simpsons were drawn on computer (I don’t know if it is or not), Homer would be a vector file.
It’s difficult to create photorealistic images, like a natural-looking face with all its shades and contours, using vector art. In fact, the few examples I’ve seen of such work resulted in a vector file as large as its raster counterpart would have been.
Which brings us to raster images. They’re used for highly-detailed images, like photographs — images with complex and subtle variations in color, light, and line. While the thousands of pixels in raster images makes them highly detailed, it’s also constraining. Enlarging a raster image beyond its 100% size will result in pixelation: that blocky, fuzzy look.