As one who works in the contracting field (and has also worked in the field, contracting), my ears perk up at the mention of “a couple of changes”.
“Change orders” are how some unscrupulous contractors make their entire living. They lowball a project estimate knowing that changes are inevitable, then once on the job the cost of changes skyrockets. (This is why one good method of contracting is to include “unit pricing” for extra work, such as a unit price for every likely material item to be used in the process. This is common in industrial contracting, but not so much in residential and small commercial projects.)
But yours doesn’t sound that way (because you would have been hit from the start of the first project, more than likely). This sounds like an honest mistake on the “shed project”, probably stemming from the original order. What were the changes, specifically (this is a rhetorical question; I’m not looking for you to post a list of changes here), and how were the changes estimated and priced?
If you think the shed “as built” is worth the $4500 in materials that is being claimed, then it’s probably worth the invoiced cost, which includes the contractor’s profit. It may very well be that he underestimated the cost of some materials, and it appears that he has agreed to “eat” the difference between his estimate and the actual cost, which is very commendable of him. I’d hire that contractor.
In order to be fair, and to preserve a relationship for future business – because honest and competent contractors are worth having good relationships with – I would ask him to share the basis for his original bid of $2000 for the shed, and then discuss the changes you ordered, too, and how those were priced. Perhaps you can agree that even though the original estimate was in error against him, and he has honorably agreed to hold that price to you, you should pay the full cost of “the changes”, whatever they were. (It’s hard to see how “a few changes” could double and then triple the cost of such a small contract, but you might be surprised, too.)
But it gets down to the contract itself, how changes are to be estimated and agreed-upon, and whether the shed was bid “hard dollar” or “time and materials”. I’d try to work something out that you can both walk away from without a bad taste in your mouth about “how that guy does business”.
EDIT: Welcome to Fluther. This is an interesting first question, for sure!