Named pipes (fifo) have
four three advantages I can think of:
(Updated, thanks to feedback from Stephane Chazelas)
So one immediately obvious task you cannot achieve with an unnamed pipe is a conventional client/server application.
The last (stricken) point above about unidirectional pipes is relevant on Linux, POSIX (see
popen()) says that a pipe need only be readable or writeable, on Linux they are unidirectional. See Understanding The Linux Kernel (3rd Ed. O'Reilly) for Linux-specific details (p787). Other OS's offer bidirectional (unnamed) pipes.
As an example, Nagios uses a fifo for its command file. Various external processes (CGI scripts, external checks, NRPE etc) write commands/updates to this fifo and these are processed by the persistent Nagios process.
Named pipes have features not unlike TCP connections, but there are important differences. Because a fifo has a persistent filesystem name you can write to it even when there is no reader, admittedly the writes will block (without async or non-blocking I/O), though you won't loose data if the receiver isn't started (or is being restarted).
For reference, see also Unix domain sockets, and the answer to this Stackoverflow question which summarises the main IPC methods, and this one which talks about