Mildred and Montag's marriage has been in trouble for years. While she recedes deeper and deeper in to the world of her television without substance, he is moving away from the society and towards individualism. When he finally reaches his breaking point, Montag is in desperate need of a confidant, a person with whom he can talk about his feeling honestly. Clarisse is gone, so he has nobody but Mildred left.
"Whether we like this or not, we're in it. I've never asked for much from you in all these years, but I ask it now, I plead for it. We've got to start somewhere here, figuring out why we're in such a mess, you and the medicine at night, and the car, and me and my work. We're heading right for the cliff, Millie."
Mildred, however, doesn't want anything to do with individualism or books. She is perfectly happy to remain in her false world, populated by false people and false emotion. The real world scares her; she has no coping mechanisms to deal with real life and real problems. Instead of helping Montag and running away from the city with him, she turns him in. Despite this, Montag finds that he can't hate her, but only feel pity.
She ran past with her body stiff, her face floured with powder, her mouth gone, without lipstick... she shoved the valise in the waiting beetle, climbed in, and sat mumbling, "Poor family, poor family, oh everything gone, everything, everything gone now...."
(Bradbury, Fahrenheit 451, Google Books)
Mildred "runs" for Beatty, because she wants her old, safe life back. She refuses to "run" for Montag because he represents change, and she doesn't want change, just the same thing day after day. Mildred's concern is entirely for herself, and she doesn't understand that Montag needs something outside of her narrow worldview. Instead, she flees to Beatty's world, which she believes to be safe, and is killed when the city is bombed.