The quotation "a merry war betwixt" refers to the way Benedict and Beatrice fight a duel of words each time they meet.
Both insult one another through jokes and puns, in an intelligent way, using irony and sarcasm, that have the objective to irritate the opponent. This competition seems to amuse them.
Benedict and Beatrice look like they hate one another, but Shakespeare uses his talent to show the psychological aspects of these characters, making the audience notice that, although the young couple cannot see it, they love each other.