Fringillidae vs Emberizidae

New World Sparrows aren't actually sparrows, but are close to buntings. Conversely, New World Buntings are not true buntings, but are close to cardinals. Darwin’s Finches and Latin American Honeycreepers are tanagers; Hawaiian Honeycreepers, Asian Grosbeaks, nearctic Evening Grosbeak and holarctic Pine Grosbeak are finches; Snowfinches are sparrows.

Up to the 1960s Fringillidae (Grosbeaks, Finches, New World Sparrows, and Buntings),
next to Fringillinae (Chaffinch and Brambling) and Carduelinae (Purple Finches, Goldfinches, and Allies), also included the subfamilies

Richmondeninae (Cardinals and Allies): Cardinals, Grosbeaks, New World Buntings, (now in Cardinalidae),
Emberizinae (New World Sparrows and Old World Buntings, the latter now in Emberizidae):
New World Sparrows, Lark Bunting, Brushfinches, finches (now in Passerellidae),
Longspurs and Snow Buntings (now in Calcariidae),
Grassquits, Seedeaters, finches,
and Geospizinae (Ground Finches): Galápagos or Darwin’s Finches (now in Thraupidae),

but nowadays only the true Finches.

During the 1960s, the Emberizidae family was inserted, containing the subfamilies Emberizinae, Cardinalinae (sometimes Pyrrhuloxiinae), and (ex-Thraupidae) Thraupinae (with the Neotropical honeycreepers sometimes in Coerebinae).

Only by the early 1980s, Clements, and the American Ornithologist Union recognized the family Emberizidae to be listed after the vireos, and (temporarily) included the subfamilies wood warblers (Parulinae), Bananaquit (Coerebinae), tanagers (Thraupinae), cardinal grosbeaks and New World buntings (Cardinalinae), Old World buntings and New World sparrows (Emberizinae), and blackbirds (Icterinae).

Howard & Moore in 1980 listed Emberizidae with subfamilies Emberizinae (Buntings), Cardinalinae (Cardinal-Grosbeaks), Catamblyrhynchinae (Plush capped Finch), Tersininae (Swallow Tanager) and Thraupinae (Tanagers). By 2003, Emberizinae was suspended, leaving Emberizidae, while Cardinalinae became Cardinalidae, and the latter three Thraupidae. By 2013, Passerellidae and Calcariidae were split from Emberizidae, and Mitrospingidae, Phaenicophilidae, Rhodinocichlidae and Calyptophilidae from Thraupidae.

So, only Fringillinae and Carduelinae were left in Fringillidae, with the Geospizinae transferred as a tribe to Emberizinae in Emberizidae (and later to Thraupidae). In North American check lists, Drepanidinae became also a subfamily, but in 2013 came under Carduelinae. Drepanidinae (Hawaiian Finches) and Geospizinae (Galapagos Finches) seem not to have been Fringillidae subfamilies simultaneously.

Country wise, China has the most Emberizidae and the most Fringillidae species.

Family ends with ~idae
Subfamily ends with ~inae

Sources: The Howard and Moore complete checklist of the birds of the world, Wikipedia

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