I agree with dlanod, but disagree somewhat with Yaztromo.
In general Tolkien's writing certainly was racist and heavily concerned with "purity of blood", but I don't think it's relevant in this instance. The line of the Stewards of Gondor is described as being "pure" Numenorean, second only to the Kings. Pippin reflects on how strongly Denethor's demeanour and appearance remind him of Aragorn. I don't think Aragorn's "higher" blood and long-ago elvish heritage make much difference -- besides, it's entirely possible that the Stewards are also remote descendants of Elros.
I think the clue is in Aragorn's remark that it was "a sore trial for Boromir, a warrior, and a lord of Men." (At the beginning of The Two Towers, IIRC.) From birth, Boromir had believed he would one day rule the men of Gondor. He was trained as a leader and used to exercising power on his father's behalf. All of this fed his natural pride and left him vulnerable to the corruption of the Ring.
Of course, Aragon was also a warrior and leader, but he was not only that. He had spent much of his life wandering alone, hiding his true heritage. The only people he had authority over were the handful of surviving Dunedain in Arnor. The people of Bree seem to have thought of him as a highly suspicious outsider. Last but not least, he was raised and educated by the Elves of Rivendell, who would have had an entirely different perspective on power from that of Denethor. All this would have made him much better able to resist the Ring.
Similarly, Faramir never thought of himself as destined to rule. He is described as more bookish and intellectual than Boromir. So his mindset may have more closely resembled Aragorn's and allowed him, too, to resist the temptation of the Ring.
Another factor is that Boromir had to face the prospect of Aragorn becoming King of Gondor. Instead of being a ruler as he had always expected, he would be only Aragorn's subject. (IIRC this is made much more explicit in the films than the book.) One way Boromir could avoid this was by seizing the Ring, defeating Sauron and setting himself up as a king. This would certainly have been part of his motivation.