I read in the various reloading manuals the different loads and the associated cases. What is the difference between say, a Remington case and a Winchester? Is it important that I only use the case shown? What will happen if I mix and match them?
I' ve always tried to keep them together but sometimes I have run out of one and finished off with another brand.
I' ve read that military brass is heavier and therefore smaller inside, but never really anything on civilian brass.

If you weigh your brass you will find it varies even if it is all done by the same manufacturer. By mixing brass You shouldn' t have any trouble with it other than a difference of burn rate as long as your not loading Max loads . Accuracy could suffer a bit .

Don' t mix your brass.
You can use any brand you want, they are just showing you what they used. Just remember to start at least a couple grains below the maximums listed in the books and then stick to the same brand of brass thereafter. If you switch brass you should start over working up your powder charge.

Changing any component of a load will change the performance of the load.
Need to use the same brass, primer, powder, powder charge, and bullet seating depth every time or the velocity or point of impact will change with each component change you make.

If you are shooting max loads this is very important as mixing your brass could result in some dangerous pressures.
At the least you will experience point of impact changes from using different brands of brass, all other components being equal.

If you are shooting groups or trying to sight in with mixed brass lots this will really mess you up.