Borromini accused Bernini of exploiting his (Borromini's) skills and knowledge to become the favored architect of Rome's "great families." In fact, both Borromini and Bernini sought and received high level Roman patronage, and the fact that they were appealing to the same rich families made their rivalry even more bitter. It was said that Bernini finally nominated Borromini as architect for the proposed church of S. Carlo alle Quattro Fontane to be rid of him at S. Pietro. Borromini took up the challenge of an exceedingly cramped space to produce what is considered his masterpiece, which is still called the "Carlino" because of its small size. A few years later, Borromini extracted some small vengeance when he attracted the favor of the increasingly influential Pamphilj family and got the commission for the Palazzo Pamphilj and S. Agnese in Piazza Navona. Bernini, according to legend, had to use trickery to get the Pamphilj commission for the Fountain of the Four Rivers in the center of the Piazza.
Somewhere along the line, Borromini made an unfortunate "image" decision. Because his rival, Bernini, hobnobbed with the "great families" and dressed and acted like them, Borromini decided to go the other way and started to show up for appointments in the Palazzi dressed as a common workman, thinking he would thus ridicule Bernini's pretensions. The ploy backfired, however: the nobles accepted Bernini (who was, after all, from a noble family) and treated Borromini according to his humble attire and origins.
Borromini was, by all accounts, not a pleasant man, and he argued with most of his patrons as well as his employees. He reputedly beat one of his own assistants so brutally that the assistant died. Because of his "image" and his bad temper, commissions dried up. By 1667, Borromini was an angry, frustrated, and depressed old man, and he committed suicide in the summer of that year by impaling himself on his own sword.
After Borromini died, Bernini continued his anti-Borromini propaganda, which was taken up by the Italian architectural community. Borromini was considered by succeeding generations of architects to be too radical, excessive (as if all of Baroque is not!), and "hurtful to the eyes." Foreign architects seem to agree, transferring their displeasure with everything in Italian Baroque to Borromini's shoulders, perhaps because Bernini himself is such a cultural icon (because of his sculpture) that he is unassailable. Bernini wins.
http://www.borromini.at/e/index.html The Borromini Exhibition
http://www.siba.fi/~kkoskim/rooma/pages/FALD306B.HTM S. Agnese in Agone
http://www.bestitaly.net/morlacchi/01b.htm S. Agnese in Agone
http://www.aol-b.com/card93.htm Sant'Ivo in Sapienza
http://www.mcm.acu.edu/academic/galileo/ars/arshtml/arch4.html Portico in Palazzo Spada