It's also worth considering whether there are sales tax savings involved.
In some states (e.g. Illinois), you can sell a car privately and deduct the selling price from the car you're buying from a dealer before calculating the sales tax; this could save you thousands (perhaps $4K) either way.
In some states, you can only deduct the selling price if you're trading the car in at the same dealer where you're buying; this could save you thousands, but only as a trade-in, not in a private sale.
In some states (e.g. California, as I understand it), you have to pay sales tax on the full purchase price of the car you're buying, regardless of whether you're trading in another car or not.
And in some states (e.g. Oregon), there isn't any sales tax to worry about, so there is no tax effect on either transaction.