The difference, of course, is that under common law the courts can, do and should make a distinction between the relative powers of the contracting parties.
There is a question about whether there can be any sort of contractual relationship (in the civil sense) between a state and an individual. All states recognize that a contract signed under duress is voidable. Yet ANY contract undertaken with a state is made under duress (why else sign a contract with a state?).
And to go back to my original point, no private contractant could, under breach of contract, throw you in jail and take away your rights as a citizen. Only a state can do that.
To make any attempt at forcing these issues into a Constitutional question however... you first have to negate the contract. Until you do it's not a Constitutional issue so the correct place to start is in fact (as another FReeper above suggested) to decline their invitation to apply for a license, turn in the one you have and claim that right to travel freely by the common mode of the day.
It is at that point, you prove the contract is only in place under duress and then you have only to prove that travel is a right, then that the right is rather meaningless in today's society if it can not be done by driving a car on the public roadways. Quite a difficult and painful process, but I have seen it done. Just not very realistic today and in the meanwhile courts are in fact treating this as not a constitutional issue. While judges are known to lean on the states a bit to keep the "contract" slightly more fair than some states might like, there is nothing there that looks anything like the "secure in your persons and papers" or as having any right against self incrimination, etc... When in the context of driving and the driver's license, there are very clear infringements on the rights a Freeman might otherwise exercise and it's my personal believe after some considerable time spent studying the issue that they are able to do this by treating it as a simple contract situation.