What is the difference between a transistor being use as a switch and as an amplifier?
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This question can be more specific in sense the type of transistor we are talking about. It can be bipolar transistors or the MOSFET(s)(metal oxide semiconductor field effect transistor).

As far as my knowledge is concerned. In case of MOSFET, if we consider the three terminals; namely source , drain and gate ( considering the fourth terminal is connected to the appropriate voltage), there are many cases.

Switch can be made out of NMOS (N-channel MOSFET), PMOS (P-channel MOSFET) or both ( CMOS- complementary MOSFET, combination of NMOS and PMOS).

Basic point is this , when Vgs ( gate to source ) voltage is above threshold voltage ( Vth) and Vds ( drain to source ) voltage is greater than (Vgs-Vth) then MOSFET starts conducting properly . So for a switch applying a gate control voltage we can turn on or off the device, making it as a switch. For case of CMOS , an Inverter is best of example of switch.

Now as an amplifier a lot can't be explained through a plain simple text. But in nut shell when MOSFET is in saturation , fully turned on (Vds> Vgs-VTh and Vgs> Vth). When a small signal (ac) is applied at the gate , with constant current (dc) flowing in a MOSFET ( from drain to source or vice-versa for NMOS and PMOS respectively) the output voltage ( small signal voltage output) of MOSFET as an amplifier is gmro . Where gm is transconductance of amplifier (MOSFET) in saturation region and ro is the out put resistance of amplifier or MOSFET in saturation region.

There are several configuration of amplifiers , differential amplifiers ( Diff -amps), operational amplifiers (op-amps), etc.

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