무섭다 is a more common, everyday expression, while 두렵다 sounds somewhat more formal or literary. 무섭다 can also describe a more immediate feeling of fear, whereas 두렵다 often conveys a deeper sense of fear. 그 영화는 무서워. That movie is scary. 실패한 삶이 될까봐 두렵다. I’m afraid that my life will end up being a failure.
Would you say that 두렵다 is more existential?
Just a random aside but 두렵다, 두려워하다 and 두려움 do come up in everyday conversation sometimes. Maybe it's because of my own experiences but I use them too (I'm Korean) and I feel like I've heard people around me use them fairly often. People often use them when talking about rescue animals with trauma that are still curled up and too scared to open up to their new owners. (쟤좀봐 사람이 만지는게 두려운가봐) I also hear them used for someone who's sick, injured, about to have surgery, or just really anxious. (야 너무 두려워하면 될일도 안돼)