Spirit of America versus the law of the United States
As you may already know, the key difference between the Declaration of Independence and the Constitution/Bill of Rights is that the latter forms the basis of law. The prior is merely a dream and an idea, lending cultural inertia to how we interpret the law.
However, a key component of our ethos is the idea that we can remove a government that goes afoul of our "inalienable rights". Anyone would agree that our government is not representative except of those interests who can afford to gain the attentions of these officials.
"The meaning of "equally free and independent" is unclear and ambiguous, so say the courts of America. The "pursuit of happiness" is one of the "unalienable rights" of people enumerated in the Declaration of Independence, along with "life" and "liberty." "The right to pursue any lawful business or vocation, in any manner not inconsistent with the equal rights of others, which may increase their prosperity or develop their faculties, so as to give them their highest enjoyment." Butchers' Co. v. Crescent City Co., 111 U.S. 746, 757, (1884.)"
"Because the right is not set forth in the U. S. Constitution, it is not enforceable by the courts. However, the right to the pursuit of happiness is often raised in arguments against government regulations, because its mention in the Declaration of Independence gives it a degree of forcefulness. Barron's Law Dictionary, 2nd Ed, pg.378."
"We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness. --That to secure these rights, Governments are instituted among Men, deriving their just powers from the consent of the governed, --That whenever any Form of Government becomes destructive of these ends, it is the Right of the People to alter or to abolish it, and to institute new Government, laying its foundation on such principles and organizing its powers in such form, as to them shall seem most likely to effect their Safety and Happiness. Prudence, indeed, will dictate that Governments long established should not be changed for light and transient causes; and accordingly all experience hath shewn, that mankind are more disposed to suffer, while evils are sufferable, than to right themselves by abolishing the forms to which they are accustomed. But when a long train of abuses and usurpations, pursuing invariably the same Object evinces a design to reduce them under absolute Despotism, it is their right, it is their duty, to throw off such Government, and to provide new Guards for their future security. —Such has been the patient sufferance of these Colonies; and such is now the necessity which constrains them to alter their former Systems of Government. The history of the present King of Great Britain [George III] is a history of repeated injuries and usurpations, all having in direct object the establishment of an absolute Tyranny over these States. To prove this, let Facts be submitted to a candid world.