What is the purpose/benefit of using different electrodes in the same circuit?
You always need at least 2 electrodes (anode and cathode).  If you only have 1 electrode, you can't build a circuit.  

In electrochemistry, the electrodes generally have some role in the reaction above and beyond just supplying electrons.  This determines what materials they must be made out of.  

In the case of an LCD screen, the liquid crystal is affected by the ambient electric field.  Two electrodes produce a well-defined electric field in between them.  Basically, they end up acting like a capacitor.  In some LCD screens (e.g. TFT) the electrodes also come along with transistors patterned onto the glass for better performance.  

In measurement applications, you need two electrodes as you either need a current path or you need two electrodes in order to define the voltage, as voltages are meaningless without a reference for zero.  One of these electrodes is generally 'ground', though this is not a requirement.