빌다(bil-da, bilda)
to plead earnestly or pray to a deity
When to use
Praying to deity, Pleading in desperate situation, Very earnest request
The nuance
Used mainly when earnestly wishing or praying to gods, fate, or heaven. Can be used for very earnest or formal requests to people, but not for everyday requests.
부탁하다(bu-tak-ha-da, butakhada)
to politely request help or an action from someone
When to use
Asking friend for help, Requesting work at office, Everyday requests
The nuance
Used when politely asking someone to do something. Common in everyday conversation and suitable for both formal and informal situations.