To defend Wilkins, it was John Randall, the director of the lab Wilkins and Franklin both worked in, who probably intentionally pitted them against each other to mess with or motivate Wilkins. Wilkins was possibly the most honorable out of all five people involved in the situation.
Wilkins was "second-in-command" to Randall, developed the DNA structure project, and convinced Randall to assign more people to work on it. Randall then hired Franklin, reassigned Gosling, the graduate student who had been working with Wilkins, to Franklin, and told Franklin that Wilkins would simply be handing over his data to her and that she would subsequently have full ownership of the project. Randall didn't tell Wilkins any of this of course, so a lot of hard feelings developed between Franklin and him. The situation got worse when Wilkins tried to get sample from external collaborators to continue working on the project himself and Randall forced him to hand over one of the samples to Franklin. Franklin finally got sick of Randall herself and left, leaving Randall to turn over all the data to Wilkins, who then went to talk about his pet research interest with Crick, a personal friend of his. Wilkins then recused himself from Crick's paper, feeling he hadn't contributed enough to it. He also worried publicly to others that maybe he had been unkind and driven Franklin out, having minimal insight into Randall's tactics, which are unfortunately common in the field. When they're being used on you by someone skilled in them, it's often hard to realize, and you end up being resentful of the person you're being pitted against until one of you leaves and you suddenly have clarity because the stress of the situation is suddenly reduced.