(Speaking of, I still like The Silmarillion better, even if the split of the line of kings WAS interesting. LOTR-proper has an awful lack of delectable renegade elves. I just. That is an absolutely necessary component. :P)
Aragorn's backstory was known to me in general, but I was surprised to learn that his sojourn in distant lands included a stint as a hero of Gondor under an assumed name (Thorongil), during which he served Denethor's father. Having seen the movies most recently, which mention that Aragorn was hanging around in Rohan, at least, during Theoden's youth, I kind of assumed he was also in Gondor (he says he has seen Minas Tirith, after all), but put it from my mind because I was more interested in his relationship with Elrond. Turns out I should've looked into that, because Thorongil was incredibly popular in Gondor and of special interest to Denethor's father, where as Denethor "was ever placed second to the stranger in the hearts of men and the esteem of his father," and rather disliked Aragorn. (I can't give you page numbers since I read the Kindle version; this is around location 23268, whatever that corresponds to in pages.) Since Denethor is supposed to be extremely perceptive, it's not a stretch to suspect he knew who "Thorongil" really was. In fact, since the text goes so far as saying exactly that, I will just assume he figured it out for real.
That's a long time to be looking over your shoulder for usurpers and glaring at old men in pointy grey hats. I can't fault Denethor for his attachment to his office; I can't fault him for being proud when it's clear he is deeper-seeing than men of his time, or for resenting a man who came between him and his father's love, or seemed to. And I keep thinking about Faramir's anecdote about his brother:
‘And this I remember of Boromir as a boy, when we together learned the tale of our sires and the history of our city, that always it displeased him that his father was not king. “How many hundreds of years needs it to make a steward a king, if the king returns not?” he asked. “Few years, maybe, in other places of less royalty,” my father answered. “In Gondor ten thousand years would not suffice.” Alas! poor Boromir. Does that not tell you something of him?’ (p.669)
This says as much about Denethor as it does about Boromir. They seem alike in temperament even if they're different in every other way. And Boromir always seemed to me the character most like classic mythical or literary hero - someone with heroic qualities and a fatal flaw to overcome. Neither he nor Denethor overcome, and that's too bad, because that transformation would turn them into that paragon of men that some readers or viewers want to see, which Aragorn and Faramir are seemingly by birth. Granted Aragorn goes through his own development arc, as Samu mentioned, but even in his youth it seemed he was placed above other men. He was simply born Better.
Well. That's all to say that I know too well what the long-term effects of paranoia are, and I'll cut Denethor some slack. Also, as Mark mentioned earlier, the staging of his death in the movie lends itself to an interpretation of his character that's not quite right. The movie makes me think he's committing suicide because he's completely out of it, and the book makes me think he was deeply, maybe irreversibly depressed-- despairing would be the theme-appropriate word, sorry. That does make a big difference.
I still don't like him, though.