Digital filters are mostly used in digital communication systems. I don't know what you mean by their usage in computers.
Baseband (I/Q) signals in DSP/FPGA are filtered by digital filters.
Analog memory? That certainly isn't anything that has any mainstream practical examples. There are few things around that have tried to be analog-ish, with multi-leveled voltages storage of bits to increase the bit storage capacity of a single cap, but this is still more digital-ish than analog-ish.
The only thing that we could barely call as "analog memory" is the switched capacitors on CCD sensors, but it isn't possible to do anything else than serially shift their content.
It depends.
* with an IIR: you get phase shift. It is a continous (steady?) phase and amplitude behaviour.
* with FIR: you may create filters without phase shift (but fixed delay). They often need more processing power and more ram usage.
There are so many other things that differ. Each filter has it´s advantages and disadvantages..