If you are close, get your realtor to negotiate the difference and make it a sale. Don't loose a buyer in this market over what appears to be a small difference.
So, my advise is to get the deal done asap. Listen to your listing realtor.
Of course I now just noticed that this is an old question, so what ever happened?
The lender's appraisal will stand, and if you want to sell the home (any time in the near future), you will need to accept the value of the appraisal. A lender is NOT going to lend money for more than the appraisal (no matter what buyer comes along), and rightly so with all the trouble the market is in because of bad lending practices of the past.
This is unfortunate. However, the newer revisions of our NEFAR contract have all the types of lending needing the appraisal to be equal or higher than the purchase price.
General information regarding my experience with appraisals---Not sure when the first appraisal was taken, but as we are declining at about (roughly) 3% in some areas per month, if the first appraisal was taken over 6 months ago it would be invalid. If the first appraisal is taken within 3-6 months perhaps it can be taken into consideration--but in my experience the appraisal for the lender needs to be within 3 months.
The next issue is this---if you and the buyer do not come to terms, and the home goes back on the market, the appraisal (current) could indicate that the value of your property is in one of the greater declining areas--so the risk is a lower appraisal should another buyer come around rises.
I am unclear if you have a Realtor representing you, and if you do all of this becomes secondary and/or invalid as I would go with their professional opinion. The same would apply for the buyer. And, if you are doing this on your own I would consult a professional over going forward on your own. Please, again, consult your Realtor--and if you do not have one perhaps this is a good time to find one!