发愁 is a separable verb (离合词) — you can insert other words between 发 and 愁, e.g., 发什么愁 or 发了三天愁. It means 'to worry, to be anxious' about a specific problem, often with a sense of distress, whereas 担心 (dān xīn) is a general, non-separable verb used for any kind of concern or worry about someone or something.