They're synonyms but "cara" is more informal than rostro. Spanish is spoken in many different countries and some words are more common than others in different regions. For example, where I'm from we use "cara" almost exclusively and only use "rostro" for very formal occasions, songs and/or poems. It's sort of like "face" and "visage": "cara"="face" and "rostro"="visage".

I think when you are talking about people is the same to use "cara" or "rostro" , but when you are talking about something else you must to use "cara", for instance, "el rostro de esa persona me parece conocida" is fine. "la cara de esa persona me parece conocida" is fine. "la cara de mi hermano me resulta sospechosa" is fine. "el rostro de mi hermano me resulta sospechoso" is fine. "sobre una de las caras de la moneda hay un nùmero" is fine. "sobre uno de los rostros de la moneda hay un nùmero" is wrong. "La cara oculta de la luna" is fine. "El rostro oculto de la luna" is wrong. "Un cubo tiene 6 caras" is fine "un cubo tiene 6 rostros" is wrong

"Me suena tu cara pero no sé de qué OR .""Me suena tu cara pero no sé de dónde"

Sorry, I forgot to answer your question, rostro is more formal and means only face.. But, as Cristina said, "cara" can have other meanings like the obverse of a coin or a plane surface.of a polyhedron.

Cara is informal and rosto is formal.