What is the difference between a modem and multiplexer?

A decoder takes n inputs and produces 2^n outputs. An encoder takes 2^n inputs and produces n outputs.

A multiplexer selects one line from many lines. The inverse of selection is distribution. A demux essentially transmits data from one line line to 2^n possible output lines. The output line is determined by n select lines. In short, a multiplexer selects an input line, a demultiplexer selects an output line.

The differences between these two circuits is subtle, as far as I can tell. A demux simply selects an output line, nothing more. It's a glorified switch. A decoder takes n inputs, and uses those inputs to determine which of the 2^n output lines is high. This is the difference, I think. A decoder is designed to simply keep one line high. A demux is designed to set one output equal to the input (whether it be high, low, or a changing signal).