Pavel Petrovich doesn't like Bazarov from the get-go. As soon as Bazarov gets away from Pavel Petrovich, he starts making fun of how the old man still tries to pretty himself up like a dandy. Arkady, timid as his father Nikolai, finds himself caught between the two proud men. Yet the two are remarkably similar, which is, perhaps, the reason that they dislike each other so intensely.
Both are, above all, very proud and convinced of their own way of thinking. Though Bazarov doesn't believe in principles, his very disbelief comes to constitute a principle that he clings to as stubbornly as Pavel Petrovich clings to aristocratic tradition (if this is confusing, note that the line "There are no principles" is, itself, a principle). The strongest bond, though, might be that, by the end of the story, both Bazarov and Pavel Petrovich have been spurned by the woman they love and resolved to live without romantic love.
Perhaps their similarities are outweighed by the fact that Pavel does not like to see a young man with the stubborn vitality he once had, and Bazarov does not like to see the proud old gentry-man that he might one day become. Their situation is best described when they come to an understanding after their duel, and the narrator notes, "Such mutual recognition is agreeable between friends but most disagreeable between enemies, especially where it is impossible for them either to thrash things out or to part company" (24.94).