Clarisse's family, who lives next door to the Montag's house, is different in that they are still interested in each other as human beings. While people in the rest of society (represented primarily in Mildred) live their lives connected only on a superficial level, Clarisse's family (in particular her uncle) have discussions that last into the night. While Mildred's friends joke in all seriousness about getting rid of their children as soon as possible, Clarisse's family nurtures her and gives her advice.
Honestly, I think the neighbors are different in order to provide a foil to the Montags. Clarisse sparks Guy's awakening, and she could not do so without some background that would nurture her awareness of the real world.
In regards to Montag's feelings for Clarisse, I think he is in awe of her more than he is in love with her. If she were a few years older, then I think he could fall in love with her (provided the emotion love has not been obliterated from his emotional repertoire). However, with things as they are, he ends up feeling vaguely paternal towards her, in spite of the fact that Clarisse has more knowledge of the world than he does.
In the book, the first thing that Montag notices about the house his that "all its lights were blazing." Most other houses at night are completely dark, because everyone is watching t.v. When Montag asks about it Clarisse says that it is because it is "my mother and father and uncle sitting around, talking." Then, she says it is "like being a pedestrian, only rarer." This refers to people who go outside to go for walks, just for the heck of it (pedestrians); but sitting around talking with your family is even rarer. So, her family talks, walks, and actually spends time with each other. Why? They like it; they haven't lost sight of what is important. This is different from most in their society; Mildred rarely talks to Montag. Instead, she watches the t.v. walls. Her friends speak of family as a bothersome irritation, but that at least they can "plunk their children in school nine days out of ten" and when they are home you can "heave them into the parlor and turn the switch." Most people think that "no one in his right mind...would have children" at all.
Montag finds Clarisse and the idea of her family a breath of fresh air in a world that is devoid of any real human contact. He starts to feel, for the first time that he can remember, since being a child. This means a lot to him, as does Clarisse.