This appears to be a compatibility problem between the scanner and my USB 2.0 port.
When I unplugged the USB cable from the USB 2.0 socket on my computer and plugged it into the old USB 1.1 socket instead, the scanner started working fine, and the problem appears to be solved.
I think I must have inadvertently caused the problem when I moved my computer a few weeks ago. I disconnected all the cables, including USB devices. When I reconnected, I must have connected the scanner to my newer USB 2.0 card instead of into one of the original USB 1.1 sockets on the computer.
This review of the Epson Perfection 2450 Photo describes the solution, written by someone who had exactly the same problem. However, the reviewer had bought a new computer with only USB 2.0 ports. In this situation the solution was: "So I plugged my old USB 1.1 hub into one of my new USB 2.0 ports and then plugged the scanner into the USB 1.1 hub. Voila! It works like a charm!"