"The value" or the appraised value is what I'm assuming you're referring to since the question is asked of us as appraisers, is our estimate or opinion of what a property is worth. To determine the appraised value, we look at historical closed sales which most weight is placed on as well as pending and active sales in the immediate market area to support our opinion.
The factors include location, construction (year built, style) GLA, bed, & bath count, features, blah blah, etc. We slog through this supposedly to help the lender gauge the risk on making a sound lending decision. And.... to make sure the buyer doesn't financially "over commit" (which doesn't work BTW).
"The price" is the actual amount the property sells for to a supposedly informed buyer who is not being duped by their realtor. Again, current market conditions, demand, and how many parties do they have to out bid.
If the "appraised value" and the "contract price" differ.... this puts a big stick in the spokes and makes everybody "unhappy". Disgruntled realtor calls, stips, and ROVs ensue....
If the appraised value matches the contract price.... winner winner chicken dinner..... everybody's happy! This is "the price".
The End