First Agile is actually a number of different processes that follow a similar philosophy. One of the philosophy's that makes it different is that each iteration produces a working product. It could be described as iterative and incremental. A lot of emphasis is placed on the working product and on testing. In many agile models testing comes before coding.
In the spiral model the number of iterations are fixed, while each phase of an agile model may consist of any number of iterations.
You are right that there are similarities but the underlying philosophy makes the difference. This page explains in more detail and compares agile to other methods.
You can say that agile processes are Use Case driven...placing a lot of emphasis on people, the end user.
