Although there are a tremendous amount of differences between the two characters, there are some similarities also.
1. They are both females who live in the same neighborhood and live under the same rules, regulations, and government.
2. They both have a distant attachment to Montag. Clarisse has just met him, having just moved into the neighborhood. (pg 6) Even though Mildred and Montag have only been married ten years, Mildred can't tell him where they met or when. (pg 43). He can't remember either. It is important to him, and she says, "It doesn't matter". Bradbury tells us "And he remembered thinking then that if she died, he was certain he wouldn't cry. For it would be the dying of an unknown." (pg 44)
3. Both Clarisse and Mildred do not think they have anything to fear from the government. Clarisse sees the government people as a game. They make her go to a psychiatrist and she "makes up things to say". (pg 22) She says that "they" (the government) want to know what she does with her time. She tells them she sits and thinks, but she won't tell them what she thinks about. "I've got them running". Mildred has been brainwashed by the government and spends her days mesmerized by the three walls and the characters on it she considers "family". She turns in Montag, who will be arrested, for the books he has in the house and flees even as her house burns. She thinks if she turns him in, she is safe. She has done what the government and her peers have told her to do.
4. They are both killed by the government. Beatty tells Montag that they have been watching Clarisse's family and that "she was a time bomb." (pg 60) Her problem was that she "didn't want to know how something was done, but why. .....She is better off dead."(pg 60) Then he adds, "Luckily, queer ones like her don't happen often. We know how to nip most of them in the bud early." (pg 60) Mildred told him that Clarisse was run over by a car. (pg 47) They never come out and directly say that the government killed her, but it is heavily inferred. Mildred is killed at the end of the book by the atom bomb that is dropped during the one day war. (pg 159) Montag imagines "that at last she recognized it (her face) as her own and looked quickly up at the ceiling as it and the entire structure of the hotel blasted down upon her, carrying her with a million pounds of brick, metal, plaster, and wood...where an explosion rid itself of them in its own unreasonable way" (pg 160)