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IIR filters are recursive and FIR filters are non-recursive. Also FIR filters are linear phase and IIR filters are not; several applications are sensitive to non-linear phase (communications, medical, etc). In implementation, IIR filters require fewer taps (smaller order) and thusly are easier to implement and have fewer zeros. Also FIR filters are always stable, while IIR filters can often become unstable in implementation. The previous answer is correct about delays.

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