In an FIR filter, the lack of an input implies the lack of an output. That is the filter will not continue to produce an output in the absence of an input.
In an IIR filter, an output will continue to be produced even after the input goes away.
IIR filters involve the use of feedback and can be simpler to implement.
The IIR filters closely approximate the corresponding analog filters. Inductors and capacitors have "memory" that decays exponetially with time and can be ignored past a certain point but never becomes identically zero. The FIR filter is only possible in the digital domain, and like all digital filters the input must be band limited by an appropriate anti-aliasing filter.
I found that IIR filters have much better frequency response than FIR filters. Why exactly does that make it an advantage please? I cannot exactly understand that is meant by better frequency response.
FIR filters do not feed back the output to the input. An advantage to this is that they will not oscillate ( oscillation is almost always undesirable.)
FIR filters require more filter stages for a given attenuation than IIR filters: they are computationally expensive.