There isn't much difference. The packaging system is exactly the same, and all packages are built automatically on all architectures from the same source. (The source can contain different compilation settings, of course.) Packages are only accepted in Debian testing or Debian stable if they are available for all supported architectures (which has included ARM since potato in 2000). There are a few exceptions, by explicit dispensation — mostly “exotic” compilers and closed-source non-free programs.
There are two ARM architectures in Debian: armel (following the current ARM binary interface, EABI), and armhf (for processors with hardware floating point), which is expected to be officially released with the upcoming wheezy release. Binaries in armhf are faster for applications that use floating point, e.g. video encoding and decoding.
Debian's armhf does not work on a Raspberry Pi, because although the Pi has hardware floating point, it only supports an older version of the instruction set, but you can use the specialized distribution Raspbian, which recompiles Debian packages optimized for the Pi's processor.
For the kind of network appliance you describe (router, firewall, file server, ...), the processor architecture doesn't matter. An ARM or MIPS device will do fine.
There isn't much of a difference in RAM consumption between ARM and x86. ARM code is a more compact, but for a network appliance most of the RAM usage will be for the data that you're serving, the network packets you're relaying, etc.