Darzee and his wife are both tailorbirds. Darzee is a songbird. In other words, he's got a beautiful voice, but that's about it. In practical matters, his mind is too prone to flights of fancy, as we see when he composes songs for Rikki-tikki's victory while not doing anything else to help the mongoose (64-70).
Darzee's wife is the exact opposite of her husband. Practical and levelheaded, she knows she must help Rikki-tikki by distracting Nagaina, so the mongoose can destroy the cobra nest. She also warns Rikki-tikki about Nagaina's attack on the veranda and distracts Nagaina again so the mongoose can bite her, ensuring his victory.
In a way, both birds take on the role of foil for Rikki-tikki. Darzee is a flighty artist versus a manly bird of action, which lets us see the advantages to Rikki-tikki's more action-oriented personality. Meanwhile, Darzee's wife understands the need for action like Rikki-tikki, but unlike the mongoose, she lacks the natural ability to do much.
This character analysis leaves us with two questions:
(1) Why does Kipling make Teddy's parents stereotypically gendered—practical dude, flighty lady who faints at the first sign of trouble—while showing the native birds the other way around? Is it a comment on Indian gender relationships? On civilization versus nature? Are we reading too much into it? (Never!)
(2) Why would a writer create a character like Darzee and then point out how pointless, perhaps even stupid, his art is in the world? Does that feel weird to anyone else?