A zero represents differentiation of the input signal, while a pole integrates it. From a control theory perspective, a zero makes for a 'PD' controller which, like the proverbial hare, responds fast but with steady state error. OTOH a pole corresponds to a 'PI' controller that, like the tortoise, is slow to get there but does so with zero steady state error.

Here's a physical picture: say you're designing an elevator for a high rise building. You leverage a zero to get quick response but it won't do for the elevator to stop three feet short of the opening to a particular floor. Likewise you use poles for a slow but precise 'Driving Miss Daisy' ride that will align the door to the opening with millimetric precision (and leave passengers wishing they'd taken the stairs). The golden mean corresponds to a balance between haste and precision - it's called a PID controller and is non-trivial to design.

Non-linear circuits can be very different. A flip-flop regenerating its own input is like a snake chasing its tail. The unstable transient response takes off like a rocket, reaching steady state nirvana by saturating against one of the rails. If you design fighter aircraft you want 'em to whip around like that, but also not spin out of control or make Top Gun pass out. So you do an elaborate balance of linear and non-linear control. Even PID design is kid stuff compared to non-linear aircraft control amidst turbulence and countermeasures.

M.G.Rajan
www.eecalc.com