難しいですな。いい質問です。。。頑張ります。
We usually use "interest" to describe a general or broad area, and "curious" to describe something more specific. It's case-by-case of course. In many sentences you could use either word.
Also, "curious" implies lack of knowledge - you don't know something, and want to know. "Interest" could mean you already know a lot about something (eg. if it is your hobby).
例えば:
I have an interest in soccer. (サッカーは趣味です。)
I am interested in movies. (映画が好きです。)
だけど:
I am curious as to where he is now. (彼はどこかな?分かりたいです。)
I am curious about how electricity works.
よく、ほとんど同じ意味です。例えば:
I am curious about biology; I want to study it next year.
I am interested in biology; I want to study it next year.
I hope that helps!