If you substitute the word dream for happiness, I think you'll see the difference. We want to live in a place that guarantees you the right to have (pursue) dreams but not mandate what those dreams will be. The only way to guarantee the right to happiness is if the government defines what happiness is--and that would be a disaster.
Happiness is an abstract noun. As such, it would be hard to support and defend the right of people to be happy. On the other hand the pursuit of that which makes us happy is much more easily defined.
What is the pursuit of happiness?
Have any of your friends ever said to you, "Nobody said life is supposed to be fair." ? Same thing as saying you have the right to pursue happiness, but not the right to happiness itself. Everybody should be free to pursue happiness, but not everyone will achieve happiness. What is happiness? Pleasure and happiness are not the same thing. Happiness is such things as being able to provide a home for your family, such things as being pretty sure that if you develop your property to make a living, or work for wages to make a living, neither the government nor some outlaw will take your living from you. The government, then should be so formed that you don't have to fear that it will use its power to oppress you, but at the same time it should be stong enough to prevent outlaws and foreign powers from oppressing you. Under such a government, you can pursue happiness.
Quite simply, it is impossible to guarantee happiness to people. As an abstract quality, it has different meaning to different people and must be achieved through struggle. There have been many a novel or story written about Utopias in which all the people of a society are "happy." However, the doling out of happiness never quite works. In Ursula LeGuin's "The Ones Who Walk Away from Omelas," for example, there must be a scapegoat to suffer since qualities only exist as the counter of another quality. The inhabitants of Omelas have their happiness dependent upon the unhappiness of one miserble child. In Aldous Huxley's Brave New World, happiness loses its effect because the inhabitants of the New World have no unhappiness as they take the drug soma to dispel any discomfiting feelings.
The word Utopia comes from Sir Thomas Moore's book, Utopia, [literally, nowheresville] in which there is a republic in which all social conflict and distress have been overcome. Individual freedom and Equality exist in a society governed by Reason. But, the flaw in Moore's book is that he does not explain How this Utopia is accomplished. Incidentally, Moore's work advocates socialism; the United States may be moving towards this ideology, but it was certainly now founded as such. The founders of the Constitution designed America to be a republic. The "American Dream" was something one worked for without having anything guaranteed.
It is the element of guarantee which differentiates the pursuit from actual end point. I believe the framers understood freedom as a process as opposed to a direct product. The framers had analyzed freedom as a concept from its Classical origins with Greece and Rome. In these settings, freedom meant political participation. To not engage in political processes meant one was enslaved. Freedom was associated with political happiness and the end product was guaranteed, to a great extent. It was also reflective of the homogeneous social orders that were present in these settings. As the expression of individual consciousness became increasingly varied and complex, the definition of freedom was altered to reflect such conditions. People were able to choose from political expressions of their autonomy to commerce based notions and other forms of the good. The framers realized that if they said "happiness," it might limit individuals' pursuit of freedom ends. The "pursuit of happiness" implies that whatever path is selected by the individual, it is their choice and on their shoulders to find happiness or find meaning in it, should happiness not be present. The Framers were ahead of their time in my mind in their understanding that freedom is a means to an end and its journey is the only element that political orders can guarantee. To do so otherwise moves the political establishment towards despotism and tyranny, ends that were the ultimate fears of the framers.
It's also important to remember just how independent in mind and spirit American culture had become by the 1770s. Actually, into the modern day it is a key component of American identity and culture that we look after ourselves, that we work hard and try to get ahead as best we are able to. As a society, we value work and hate taxes. We connect work and morality even, as did many of the people who lived in that time.
So by including merely the pursuit of happiness, perhaps the Framers meant to enshrine that personal independence and responsibility into our founding documents, to limit government's ability to limit us and our potential. To suggest that we somehow had a right to actual happiness also goes against many of the religious beliefs common to America at that time - that God helped those who helped themselves.
Thomas Jefferson named three 'Unalienable rights' ( not 'inalienable') Life, Liberty, and the Pursuit of Happiness. Jefferson chose every word of the document with specific intent, therefore most historians conclude that he chose the word 'pursuit' to indicate that happiness would be a possibility in this country for those who sought and worked for it, not a guarantee. Jefferson believed in the ingenuity of man and that each man would make his own happiness whatever that might be. Now if a man chose laziness Jefferson would argue that it was his own choice, at the same time even if a man worked hard everyday there was still a chance he might fail at his endeavor. If that occurred Jefferson's philosophy would be that each man had the individual freedom to accept failure or try again. Self-reliance was essential in a free society, not government guarantees.
The reason for this is that happiness is not something that any government can guarantee while the pursuit of happiness is something that can be guaranteed.
If a government tried to guarantee happiness itself it would have no chance of fulfilling that guarantee. There would be no way that a government could ensure that all people would be materially well off, let alone emotionally happy.
But when it comes to pursuit of happiness, that is something that can be guaranteed. The government can guarantee that people will have the chance to try to buy property and keep it. It can guarantee that people will have all sorts of rights that will allow them to pursue their happiness (like the right to travel, the right to marry who you want, etc).
I've never spoken with the "framers" so I can't tell you for sure, but this is my guess:
If everyone simply had the right to happiness, that would suggest in some way that the government could provide that happiness and that everyone should have it.
By phrasing it as the "pursuit of happiness," it implies that there must still be a rather large responsibility on each citizen in order to work to obtain that happiness and also define for themselves what happiness is.
That simple change of phrasing makes a rather large difference in what the article means.