Value is not a property of a house in the way that floor area is. It is a statement about what would probably happen if the house were exposed to a market, and reasonable people constructing that statement from the same evidence will arrive at different numbers.
The comparison method
Residential valuation almost always proceeds by comparison. Find recent sales of properties similar enough to the subject, adjust each for the ways in which it differs, and take a view on where the adjusted figures cluster. The method is simple to state and its whole difficulty lies in two words: similar and adjust.
Similarity is judged on location first, then on the attributes that the local market actually prices: size, age, condition, lot, layout, and whatever the neighbourhood treats as significant. A comparable from a different street pattern or a different building era may be physically alike and still be drawn from a different queue of buyers.
Adjustment is where judgement lives
Once comparables are chosen, each is adjusted toward the subject: if the comparable had an extra bathroom, its price is reduced by what the market appears to pay for one, and so on. The adjustment figures are themselves estimates, derived where possible from paired sales and otherwise from experience.
This is why heavily adjusted comparables are weak evidence. Each adjustment adds uncertainty, and a comparable requiring many of them is really a different property with a number attached.
Appraisal has a client and a purpose
An appraisal prepared in connection with a loan is not an answer to the question the buyer is asking. Its client is the lender, and its purpose is to establish that the security is adequate. That purpose shapes it: it favours defensible, recent, closed evidence, it treats unusual features conservatively, and it is written to be reviewed.
A buyer may rationally pay more than an appraisal supports, because a buyer is allowed to have reasons that a lender may not consider. The difference is not a mistake by either party.
Why the market price can be elsewhere
Appraisal looks backwards, at sales that have already closed. In a market that is moving, closed evidence is by construction several weeks or months old, so appraisals lag both rises and falls. This is a known property of the method rather than a flaw in any particular report.
Additionally, a price is set by the most motivated remaining buyer, whereas an appraisal describes the typical one. When motivation is unusually concentrated, in a scarce property type or a constrained neighbourhood, the gap widens.
Assessment is a third thing
The value used for property taxation is produced by a different process, on a different cycle, for a different purpose, often using mass appraisal techniques applied to whole classes of property at once. It should not be expected to equal either an appraisal or a price, and treating it as a market opinion is a common category error.