To answer this, we need to differentiate between different groups of the colonists. Not all of the colonists, of course, fought for the same ideals or reasons. All we can do is to generalize about why broad groups fought.
The Declaration of Independence was a truer form of the ideals that many of the common people fought for. The Declaration was in line with things like the writings of Thomas Paine, which were highly influential in persuading many colonists to support the revolution. Both Jefferson and Paine emphasized the idea of equality. Paine's work also emphasized the idea that monarchy was an immoral system. This idea was at least implicit in the Declaration. Paine and Jefferson set out a very democratic, populist, and egalitarian vision. This is the sort of thing that motivated many of the common people to fight.
By contrast, the Constitution was a truer form of the ideals that motivated many of the major leaders of the rebellion. These people (many of whom, like Alexander Hamilton, became Federalists) did not trust the common people. They wanted elites to rule the country, but they wanted those elites to be American elites rather than British elites. These people feared that the common people, if given control over government, would act unwisely and cause the government to do things that would hurt the country. To them, the Constitution was superior to the Declaration because it set in place a system of government that would reduce the clout of the common people.