The short answer is yes it would be different. I mean absolutely. This is because our culture is different then it was when it was first written. The document is a product of and a representative of the culture and mindset of the times. The definition of culture is the quality in a person or society that arises from a concern for what is regarded as excellent in arts, letters, manners, and scholarly pursuits. In other words culture is the evolved human capacity to classify and represent our experiences that we have had as people together in a community (which can vary from a family, all the way to a nation, and even encompass humanity as a species).
So it is exceptionally difficult if not impossible to quantify how it would be different since we have ourselves been influenced by the culture of bygone Americans and have even been influenced by the constitution itself. Furthermore, we have influenced the document in return via amendments, which has in turn influenced our current culture and mindset. However, it was born from Americans and interestingly enough there is a strong chance that even if the constitution was written today it may be similar enough to be nearly the exact same document that it is now.
But nevertheless, it would likely be different, and not only because scientific discoveries are more advanced. Politics are different now, and it may not even be written. There may be too much political pettiness, infighting and bickering to ever reach an agreement on a constitution. If written today it very well could be so massive that it would be almost unusable and impracticable. An unwieldy and monstrous creation because instead of just getting to the point, they would try to cover every possible American group—tailor it to cover the lowliest beggar all the way to the CEO of a massive corporation. Politics is a career for many now and many are very involved with it, rather then it being the civil service it was intended for.
Regardless, the spirit of the document could very well be preserved in a contemporary genesis of the American Constitution. New science would need to be taken into account, but I believe our rights would be preserved. We value freedom above all else and the pursuit of happiness, so a scientist would have a right to research any subject as long as it did not infringe on another’s freedom and pursuit of happiness (Now I understand the concept of “freedom” and “happiness” could be a can of worms in itself due to relevant interpretation, but lets just keep it simple. Ah, sometimes I just lament our current society, even if it is what makes us great).
Freedom of the press does apply to scientific publications, as it is difficult to predict how the information is used and as long as the scientist received credit for the discovery. For example, a man learns about sodium hydroxide: “Wow sodium hydroxide is awesome! I can use it for making paper, soaps and detergents and even in the production of drinking water! Wow it works great as a drain cleaner!” Or the man could say: “Wow sodium hydroxide is awesome!” I can use it as a chemical weapon or to dissolve aluminum locks and planes! Wa ha ha ha ha!” Nuclear weapons and nuclear power is another example. Same thing. Have to keep with freedom, and the consequences that go with it.
As for our own biochemistry, such as our DNA, that is our property and our privacy and would be preserved in the constitution. People from both sides of the political spectrum believe this, if for different reasons. I can’t see it as being any different. I also do not see them quarantining anyone with AIDs, which is not like other diseases that are much more contagious and deadly. Our government and society has not progressed to that stage yet. Now as for the violent, mentally ill murderer who could be reliably cured with modern pharmaceutical chemistry or genetic (biochemical) therapy having a constitutional right to refuse treatment? I’m not sure. A mentally ill person definitely, but a person who is a murderer? I suppose they would have the right to a certain degree, though one with freedom not being an option without it. The situation would likely be, “Hey you can take this cure or you can be put to death or stay in prison for the rest of you life. Your choice.”
In the end while the specifics of the document would be different, the American spirit of freedom would likely be preserved.