The story would have been very different in that Duryodhan would have had no archers to call exclusively his own, and Karna would have been a lot less mahaan than Yudisthir was, so the Pandavas would have been a lot better served. Karna would have won Draupadi, maybe shared her, and then gone on to rule Indraprastha.
One thing that may have been different - the Pandavas (excepting Yudisthir) may have urged Karna to wage war against Hastinapur during the Rajasuya yagna, and Karna, Arjun & Krishna b/w them could have subjugated Hastinapur & dethroned Dhritarashtra. Duryodhan would have thrown a challenge & gotten killed by Bhima, but the other Kauravas may have accepted Pandava rule out of fear. Dhritarashtra & Gandhari would have retired to the forest (w/o Kunti) and Bheeshma, Vidura, Drona & Kripa would have been members of Karna's court, along w/ the Pandavas.
They'd have lived happily ever after