What's the difference between using SPICE and just putting things on breadboard and testing?
SPICE is a design tool. Before SPICE, some circuits (maybe certain types of analogue filter for example) needed laborious calculations to get anywhere near the right values and then, after protoyping you'd probably adjust a few values because those laborious calculations were virtually impossible to get right (or needed several/plenty tedious iterations).

SPICE, as I use it, probably saves me, on average 1 iteration of circuit board on the  designs that I do.

I still scheme up what I think is correct. I then use spice to tweak values and probably add a few bits and pieces of circuit because I hadn't adequately considered this or that. I try not to use spice as a pre-requisite for good thinking although now and then I've started with no real thoughts on how to crack a problem and just developed something in parts and linked them together. Probably not the best way to do things but sometimes it works.

I always run spice before building anything and I can vouch for it saving my company a shed load of money each year. I probably design 20 PCBs per annum and saving one iteration per design makes it totally cost effective both in external artwork costs and the labour cost of building that extra (now not required) PCB iteration.

On the face of it spice can't do everything but as you progress, you learn how to import models, create models and apply work-arounds for weaknesses. I would not be without it.