Raster is typically used for photos or more complex realistic looking visuals. Vector is used for clean logos, cartoons, etc. At the hands of a skilled artist, Vector can look very realistic and almost like a photograph of a real subject matter but I typically associate it with sharp clean lines ideal for illustration, cartoons, logos, fonts, etc. Both have distinct styles and feel if you ask me. I am a vector junkie myself, I live in Illustrator. I love the flexibility it offers over raster.
Vector is math based where as raster is grid based and absolute. Meaning, a vector image is comprised of many coordinates like a connect the dots. There might be angle information showing how the line travels from dot to dot (or point to point), color, and stroke information as well. You can infinitely scale a vector image cause the program is mathematically changing the coordinates to be further apart proportionately. The image doesn't degrade. Raster images like JPGS, etc. are grid based. If you have a 16px x 16px (256 total squares) raster image and enlarge it to 400x x 400px (160,000 squares!) the software has to guess and make up what the other pixels (it has to guess what the other 159,744 squares might look like all based on the data of 256 squares!) might look like so the image becomes pixelated and looks like crap cause Photoshop didn't have enough information to go by to go from small to large.
Industry Standard software:
Adobe Illustrator = Vector (has raster effects, though)
Adobe Photoshop - Raster (has some vector features)