Karan Shishoo's answer is almost certainly correct (Bruce tried to kill Ra's, and only mistaken identity caused him to drag Liam Neeson's character to safety. Had he known who Ra's really was, he would not have done so), but there is an important element of the movie that that answer leaves out: the difference between personal identity and symbolic identity. Batman as a symbol and an idea, more than as a specific person, is referenced frequently in the film.
Bruce's aim in burning down the monastery was primarily to break the League of Shadows, with which he had come to disagree. Whether or not he wanted Liam Neeson's character dead, he certainly wanted Ra's al Ghul, the purportedly immortal leader and director of that organization, to be finished. That Ra's true identity was hidden behind a dupe strongly parallels Bruce's own dual identity as Bruce Wayne and Batman.
Ra's may or may not have cared much if Bruce Wayne lived or died. But he cared very much about whether or not Batman lived or died, because Batman was directly opposing his efforts. Ra's tried to leave Batman to die in the fire, and had identified the correct person "under the mask" to target for that. He wanted to kill the symbol that Bruce had made Batman into, because Bruce was using that symbol to keep Gotham together while Ra's and the League were trying to destroy it.
That is pretty symmetrical to Bruce's actions: he tried to leave Ra's to die in a fire. But Ra's' mask was more effective than Bruce's, so the result was that some guy died but the symbol of Ra's al Ghul did not. Consequently the League's plans for Gotham continued, despite setbacks.