Difference between Multiplexer/Demultiplexer and Analog Switch
If you are referring to a digital mux/demux, then the flow of information is in only one direction, from the input to the output, and the signals are strictly digital. This means that exact voltage levels are not preserved, just logic states. A basic analog switch is SPST, so it can connect two signals together and the flow of information can be in either direction. The resistance of the analog switch is relatively high when the switch is "open" and relatively low when the switch is "closed". An analog mux/demux is basically a collection of N analog switches where one end of all of the switches is connected to a common point. Some digital logic is used to decode the selection inputs and make sure that only one of the N analog switches is closed at any given time. If you set N to be 2, then the analog mux/demux can also be considered an analog SPDT switch. If you set N to four you have an SP4T switch or a 4:1 mux/demux, and so on...
Analog switches usually have a much lower on-resistance, to less than 1 Ω, and can switch currents up to a couple of hundred milliamperes. HCMOS mux/demux ICs are designed to switch signals only, and then a resistance of a couple of hundreds of ohms is often acceptable.
The definition of the multiplexer is pretty wide - it can be analog, in which case it's very similar to an analog switch; but it can also be entirely digital, built from logic gates. I assume you are asking about analog multiplexers.
The general difference between a mux/demux and an analog switch is this: a mux is a signal selector, allowing you to route a signal from N inputs to 1 output. A demux will do the opposite, routing a signal to one of the N outputs.
Analog switches, on the other hand, have topologies similar to those of regular switches: SPST, SPDT, etc.
If, for instance, you take a quad SPDT analog switch, and connect the common terminals to each other, you will create a what is topologically similar to a 4:1 multiplexer.
There are many other differences, like directionality of the signal, of course.