the values you see online from sites like Trulia, et al, are not appraisals, just estimates. The accuracy of those estimates can vary pretty widely. They are usually automatic, and only as good as the data they're using to make the estimates. I've seen these values off far more than in your case. They have not historically been very accurate in and around Chicago.
In terms of the appraisal, if the appraiser missed rooms you can certainly ask them to amend their appraisal. Accidents do happen. It may be that the appraisal was correct, and the basement beds & bath should not be counted. That varies depending on a lot of factors like egress points (to escape a fire), ceiling height, etc.
Have your agent contact the buyer's agent to ask for the appraisal to be amended if need be. You should also ask your agent to prepare a comparative market analysis that demonstrates why the property is worth more, which you can present to the appraiser with your appeal.
If the appraisal is accurate, you have a couple options. Ask the buyer to come up with the cash difference or lower your price. If the buyer has a mortgage contingency and they can't or won't make up the difference, they may simply walk away.... more
Consider the current value of your home compared to what you still owe on it. If you owe more than the home is worth, it might make sense to rent for a few years. You just need to ask yourself ~ are you comfortable being a landlord?