I think they have had different effects. The Declaration of Independence, to the extent that it articulated a vision of what rights people ought to have, was largely aspirational. The idea that men were created equal was roundly mocked on grounds of class, race and gender- not so much in the Revolutionary era, but in debates over slavery in the nineteenth century. But it took a life of its own, if you will, and some of its loftiest statements became almost normative- something we ought to believe as Americans.
The Constitution, on the other hand, was a profoundly conservative document in its own time, intended to rein in the excesses of democracy, which the Framers almost to a man saw as an evil. There is very little in it that could be interpreted as establishing freedoms that were not already enjoyed (habeas corpus, for instance). Yet its language was vague enough that it, too could be interpreted as a means by which basic liberties could be protected (for example, the Interstate Commerce Clause has been used to prevent racial discrimination and union-busting.) And it included a mechanism by which future generations could adapt it to their needs.
If there is any document that deserves to be called a Charter of Freedom, though, it would be the Bill of Rights, which established negative protections of basic fundamental rights while again allowing for the expansion of these rights in the future.