On screen, the quest to discover Lady Whistledown's identity is a major preoccupation for several characters, including Eloise and Queen Charlotte. In the second season, that obsession only grows, leading Eloise to form a sweet connection with a brand new character who wasn't in the books — a printing assistant named Theo who, like her, is into radical progressive politics. We also get to see Penelope's increasingly creative methods to conceal her identity as Whistledown (shout-out to Madame Delacroix for her assist!).
Penelope also faces a devastating choice when the Queen begins to suspect that Eloise is Whistledown: let her best friend face the Queen's wrath, or write a scandalous article about Eloise to prove that it can't be her. While Penelope's article exposing Eloise for spending time unchaperoned with Theo works to get the Queen off of Eloise's back, it also gives Eloise the last puzzle piece she needs to realise Penelope is Whistledown. The result is a devastating (and hopefully temporary) BFF breakup.
In the books, however, that plotline doesn't really exist. Queen Charlotte isn't even a character in the books, Eloise doesn't care nearly as much about uncovering Whistledown, and book readers don't learn Whistledown's identity until the fourth book in the series.