楼主,
Are you willing to xxxx?比较常用而且正式.
例如,结婚时,牧师常说的,Are you willing to marry him?嫁给他,你愿意吗?
Would you like to xxxx?比较常用于口语中,更多表示喜欢什么,偏向于什么.
例如,饭店,服务员常说的,Would you like to have tea or coffee?省略时,Tea or Coffee?
朋友对你说,Would you like to go with me?你愿意跟我走吗?