Prestige is high standing achieved through success or influence or wealth etc. "he wanted to achieve power and prestige" While Fame refers to 1.The state or quality of being widely honoured and acclaimed 2.Favourable public reputation cowboy...

"Prestige" implies high class, wealth and reputation whereas "fame" simply implies popularity. ie; many people know it.

Prestige: reputation or influence arising from success, achievement, rank, or other favorable attributes fame: widespread reputation, esp. of a favorable character; renown; public eminence