担心 means afraid of something to be happen. literally 担 is "bear load with shoulder", imagine you're standing on the edge of a cliff, you bearing your heart's load. 例如：我很担心明天下雨 发愁 means worry or unhappy about something. literally 愁 is "worry", imagine you have to hand your work over to your boss tomorrow, it's 2 o'clock now and you have no idea how to get start. 例如：工作太多了，感觉很发愁 Usually you could use them in the same case with or even without slightly modification.

He lost the game. I am worrying that if he is fine.这叫担心。 He lost the game. I am worrying how can I comfort him. 这叫发愁

担心 and 发愁,They are quite similar and synonym. 发愁 and 犯愁 are also synonym.