2D and 3D compared
The differences between 2D sprites and 3D models can be summarized as follows:
A sprite is an image, or—if it's animated—a series of images. Within the boundaries of its resolution (for example 64, x 64 pixels), the individual pixels make up the resulting image. This is a proven low-tech method, and it has been in use since the earliest video games. Even the first 3D games, such as Wolfenstein 3D and Doom, used sprites instead of models, as the former are easy to implement and require very few resources to render.
Because of the available memory and processing capabilities of video consoles and personal computers until the later part of the 1990s, sprites were everywhere. It wasn't until the appearance of dedicated vertex graphics processors for consumer systems from companies such as 3dfx, Nvidia, and ATI that sprites would be largely replaced by vertex (3D) models.
This is not to say that 3D models were totally new by then, of course. The technology had been in commercial use since the 1970s, when it was used for movie CGI and engineering in particular. In essence, both sprites and models are a representation of the same object; it's just that one contains more information than the other. Once rendered on the screen, the resulting image contains roughly the same amount of data. The biggest difference between sprites and models is the total amount of information that they can contain.
For a sprite, there is no side or back. A model, on the other hand, has information about every part of its surface. It can be rotated in front of a camera to obtain a rendering of each of those orientations. A sprite is thus equivalent to a single orientation of a model.