Hi Schmick
The shims are the "big washers" you refer to and used to align the diff centre with the pinion. They vary in thickness by thousands of an inch and allow for tolerances in manufacture. Even though they may look the same, chances are they vary by very small amounts. A micrometer would be the only way of accurately measuring them.
No two diff housing are exactly the same and that's why you can't just pull the centre and shims from one housing and slot them straight into another housing. Sure, they may fit, but it doesn't mean the diff will be set up correctly.
Setting up diffs is very exacting and requires certain special tools. You need a dial guage to accurately measure and set the pinion to crown wheel lash. The pionion gear has a collapsible spacer between the bearings that is designed to crush slightly as you tighten the pinion nut, to ensure the correct pre-load tension on the bearings and to set the pionion to the correct depth in the diff housing. You need to be able to lock the pinion whilst tightening the nut, and a spring scale or some measuring device to measure the amount of pre-load on the pionion bearings.
The pre-load and pionion to crown wheel lash are vital settings to ensure correct and durable service from the diff. Unless you know exactly what you are doing, and have the proper tools, it's beyond the average back yarder to tackle this job properly.
Hope this helps to explain why building diffs is a specialist job.