1. They both at one point, caught Montag's eye. He married Mildred, but later became good friends with Clarisse. So both women-at one point-interested him on some level.
2. They both were unfortunate casualities of the society that they lived in. Granted, their fates were on opposite ends of the spectrum: Mildred was the cold, miserable prototype of their stifling society, whereas it was hinted that Clarisse, because her and her family refused to become as Mildred was, was taken away or removed (if her family wasn't fortunate enough to escape first). So both women suffered in their society.
In almost every other way, Mildred and Clarisse seem to be polar opposites, which is one reason Montag finds Clarisse so refreshing and different from what he's used to. I hope that helps!
There are more ways in which Clarisse and Mildred contrast than they do compare. In fact, their differences far exceed their similarities. The most obvious similarity is that they are both female.
Another similarity is that they are both significant in Guy’s journey towards happiness and preservation of the information in books. It was Clarisse that prompted Guy to question his profession, his state of happiness and life as a whole. Guy then began to read books and think, he saw the world from a new angle and he wanted to fight for change. Mildred, on the other hand, reminded Guy of how shallow and empty the world has been turned into because of the brain controlling information conveyed by the walls to which the majority of people are slaves. Mildred’s lifestyle was a constant reminder to Guy that things needed to change and he needed to be involved in the change. Finally, Guy’s efforts to change were accelerated when Mildred prompted the authorities about Guy’s books and he had to flee and join the social outcasts out to keep books alive through memorization.
Another similarity is that both Clarisse and Mildred die and leave Guy pursuing his mission. Clarisse was run over by a car and Mildred died when an atomic bomb was dropped on the city.