The inspection found problems the appraisal never mentioned

Does this mean the appraiser got it wrong?

Usually not, because the appraiser was never asked to find what the inspector found.

Fannie Mae asks appraisers to report adverse conditions that were apparent during the visit or turned up in research. HUD describes the FHA appraiser's look at attics and crawl spaces as visual and not technically exhaustive. The appraiser's goal is a credible value and a check against lender condition standards.

The inspector's job is narrower in subject and deeper in detail. The ASHI Standard of Practice has the inspector describe the accessible systems and report the ones that are not functioning properly, significantly deficient, unsafe, or near the end of their normal useful lives.

Which findings tend to appear only in the inspection?

Problems you only find by spending time with each system. A walk through the rooms will not catch them.

Think of a furnace that runs but is poorly vented, a few windows that no longer seal, a slow drain under one sink, or an old water heater that still makes hot water. An appraiser may not see any of that as affecting value or safety. An inspector is paid to go looking.

What are the usual options once the inspection report arrives?

It depends on your contract. The common paths are asking the seller for repairs or credits, getting a specialist's opinion, accepting the house as it is, or canceling if an inspection contingency allows it.

The CFPB notes that with an inspection contingency a buyer can cancel without penalty if not satisfied with the results, and that a seller may or may not agree to pay for repairs. The ASHI standard also has the inspector recommend further evaluation by a qualified professional where needed, which is often the next step for roofs, foundations and sewer lines.

Does the appraisal need to change?

Not automatically. The appraisal reflects what the appraiser observed and researched as of its effective date.

If you renegotiate the price or add seller repairs, the lender needs the updated contract and may have questions of its own. Questions about the appraisal itself usually go through the lender, since the lender is the appraiser's client.