What is Vector Artwork?
Vector graphics use objects such as points, lines, curves, and polygons constructed in a language such as post script to construct visual graphics. By mathematically calculating the position and shapes of objects a vector graphic can be draw at any size simply by a series of formulas. The result of all of this is that if you have a vector graphic art file such as .ai, .eps you will never have to worry about your artwork looking grainy or pixelated when printed.
What is Raster Artwork?
Raster graphics is represented by a 2 dimensional grid of pixels. Each pixel represents a color. Depending on the color mode of the image and the file type the pixel can contain a variety of information. On a basic level a CMYK image will contain 4 color components, cyan, magenta, yellow, and black combining into a color. Similarly RGB is contains only Red, Green, Blue. Some raster file types include jpg, gif, png, bmp, and many others. Raster artwork is subject to the limitations of resolution or the number of pixels contained in the file. The higher the resolution the better the quality.
One pitfall that always gets people is that any commercially available vector format allows you to embed a raster image into it meaning the file can have multiple components (parts) some vector some raster. As such the raster parts can be insufficient quality.
So which one is better?
It simply depends on what you are trying to achieve. Using vector based programs such as adobe illustrator you can mix and match vector and raster components to create full color digital graphics that have vector components. This is the most common method people use since you can mix them together and not have to think about the nuts and bolts of the way the two formats work. Like anything else, use the proper tool for the job. If you develop primarily raster based artwork in programs such as photoshop it makes little sense to use something like illustrator just so you can have your text in vector. Simply output the final production image a sufficient resolution and you will get photographic quality results. If you plan on primarily working on vector style artwork then keep your work in your vector program. By using the right tool for the job you are increasing the quality of the final product and decreasing the artwork time. Keep in mind a raster file produced in illustrator can significantly increase the file size.