In DSP when you implement a filter you typically have conflicting requirements. If you have a particular transfer function you wish to approximate, FIR is one choice. It is relatively easy to synthesize, and has linear phase response, and no ringing (Finite Impulse Response is the name, after all). However, it may take many more “taps” (delay stages) to implement as compared to an IIR. Sometimes, you care about the extra resources required (making it fit into your FPGA) but more often you can not tolerate the latency of all of those delay stages.
With IIR, you typically have much lower latency because it requires many fewer stages to get the same-ish transfer function. Tradeoffs: ringing, perhaps problematic phase response, more complicated synthesis problem.
It sounds like IIR is about reusing previous calculations? But you don't actually want every previous sample to affect the result; it's more a side-effect of how the calculation is done?
With IIR, you typically have much lower latency because it requires many fewer stages to get the same-ish transfer function. Tradeoffs: ringing, perhaps problematic phase response, more complicated synthesis problem.