Process-based serving is a more traditional technology. With a process-based server, a new thread must be created to handle each HTTP request. Under heavy loads, this system can use a lot of RAM, and result in slower performance.
With the newer asynchronous servers (also called event-based or asynchronous I/O), multiple HTTP requests can be handled within one thread, which uses less RAM and may process heavier loads more efficiently.
The Apache Software Foundation notes that performance benchmarks tend to be a measure of configuration rather than the underlying quality of the server.