(How's that for an answer?)
If your town does assessments (not appraisals, assessments) at supposedly fair market value, and the house sells for 20% less than the assessment (and the transaction was arm's length), then next year's assessment should reflect the lower price.
It's important, first, that the transaction be arm's length. Sometimes parents sell homes to their children for less than market value. Or other situations arise that suggest that the sale price does not reflect actual value. In those cases, the assessment won't/shouldn't drop to the actual sales price.
Even if it was arm's length, depending on your town's policies, it might or might not drop the assessment all the way. It still might contend that some circumstances existed that caused the sales price to not reflect the true value. In that case, if you wish, you might appeal the assessment.
But let's say that the town actually does drop the assessment by 20%. The question, then, is whether next year the town will raise (or lower) the tax rate. A lot of towns (and cities and counties) are raising the tax rate to account for such drops. Example: Let's say the tax rate is $1 for every $100 of assessed value. So on a $100,000 house you'd pay $1,000 in taxes. But with the soft housing market, the value of the home falls to $80,000. The taxes paid would decline to $800. But, town-wide, that's a huge cut in revenues. So the town next year might raise the rate to $1.10 for every $100 of assessed value. In this example, your taxes would be $880--still a drop from $1,000, but not the 20% drop you were hoping for.
So your answer really depends on what the town does: First, with your actual assessment and, second, with the applicable tax rate.
Hope that helps.