In some ways, I would say that Clarisse is more important than Millie is. I think that is one reason that she is introduced first. Montag's meeting with her is one of the events that completely changes his life.
Clarisse is very important in that she is the first person who influences Montag to change. When he meets her, he is forced to start to think about his personal life and his relationships with other people. He is forced to think about the society that he lives in. This is very important because the rest of the book will focus on how these thoughts affect him and how he deals with them.
While Millie's near-death is important, it is not as important as Montag's meeting with Clarisse. I think that is why the meeting with Clarisse comes first -- it opens his eyes and affects the way he deals with subsequent events in the book.
One cannot really speak about Bradbury's motivations when dealing with the order of events in Fahrenheit 451. One can't really speak for a writer anytime, unless some external evidence--like the writer's own comments--gives one "inside" information. All one can do is deal with the results or effects of what is presented in the work. Commentators are not privy to writers' thought patterns.
Introducing Clarisse first creates a stark contrast with Millie, once both are revealed. Clarisse is, figuratively, placed opposite of or pitted against Millie for Montag's interest. Millie's shortcomings, and, thus, the shortcomings of Millie's society, are highlighted by her juxtaposition with Clarisse. Seriously, who would you rather spend time with? The 1966 (I believe) movie plays on this by having the same actress play both parts, and by having Clarisse and Montag meet again (sort of) after Montag's escape.
The introduction of both characters one right after the other creates this stark contrast. However, I don't know that the effect would be much changed if Millie had been introduced first, followed quickly by Clarisse. I don't know that the order here is significant.
Clarisse is introduced at almost the beginning of the novel. Her introduction was quite dramatic, with Montag suddenly setting eyes on her as he turns a corner. Unlike in 1984, a book that is often compared to Farenheit 451, Bradbury stirs things up before taking more time to establish his fictional world. In 1984, most of the beginning of the novel is devoted to fleshing out the world that Winston miserably wallows in. Julia, a character who might be compared to Clarisse, is not introduced until much later, and it is when she is introduced that the action picks up. In Farenheit 451, on the other hand, Clarisse is introduced at the beginning, so the conflicts in Montag's mind begin from almost the very start of the novel. Mildred, on the other hand, is very much a part of the world that Montag is unhappy with, and descriptions of Mildred would also be descriptions of that world. I guess that because it was Bradbury's intent to get Montag questioning the world he lives in from the very beginning, Clarisse is introduced before Mildred.