Covered Up: "There She Goes" is the best-known example, originally by The La's. Also "Don't Dream It's Over", originally by Crowded House.
They've done so many cover songs that the general public seems to perceive them as "that cover band who also happened to sing 'Kiss Me'."
Face of the Band: An interesting two-person example, as Matt Slocum and Leigh Nash are well recognized, while most would be hard-pressed to name another member of the band, past or present.
Harsher in Hindsight: Any song that Leigh Nash wrote about relationship troubles, including "Easy to Ignore" and "Down and Out of Time". She and her husband divorced in 2007.