Me siento bien refers to someone who has just overcome from something bad. Me encuentro bien just means you are fine :)

I would say that both have the same meaning, but as Carlos said "Me siento bien" is more used after overcoming an illness

Me siento bien is more used, when X person has just overcome an illness or injury

'Me encuentro bien' se refiere sobre todo a la salud de la persona. 'Me siento bien' se refiere más al estado de ánimo. Aunque no es incorrecto usar cualquiera de ellas en vez de la otra. -- 'Me encuentro bien' refers mainly to the health of the person. 'Me siento bien' refers more to the mood. Although it is not wrong to use either one instead of the other.