There is a fundamental mismatch when trying to compare networked publics and participatory cultures, because they are to compare interaction through different lens for different purposes. Participatory culture refers to the overall structure of interactive spaces, be they physical or digital. It deals with the ways in which individuals participate in the creation of culture through participatory spaces, and how the structure of such spaces affects an individual's ability to meaningfully participate. The concept of networked publics, on the other hand, focuses on the manner in which inidividuals choose to present themselves based on their own ability to control their social interactions.
Essentially, I'd boil it down to this: Participatory Culture is about the individual, and how they participate in certain spaces to create culture. Networked Publics is about the individual and how they choose to interact with various audiences in different spaces. The setting for both concepts is the same; partcipatory, interactive spaces. The actors are even the same: individuals interacting with each other in these spaces. But the aspect being analyzed is different, which makes comparing them akin to comparing Marxist and Feminist critiques.
Similar to Marxist and Feminst critiques, I feel that the concepts of Participatory Culture and Networked Publics should be used together, in order to achieve a more rounded understanding of the situation that has arisen. The two concpets balace each other, with different perspectives and ways of understanding the social interactions occuring around us.
The "No Filter" article can been understood through these two lenses, as it deals with socialinteraction online. The Participatory culture aspect of the article would be the use of social media to create and contribute to various intersections of cultural identity, mostly ethnically informed. The Networked Publics aspect can be seen most clearly when the article begins discussing how conflicts crop up due to the diversity of the followers being addressed (the relevant "public" in this case). There is an understanding by people posting to this public of the people that they're addressing, and that not all of them will agree, which leads to a conscious controlling of the message to fit that public.