Willingness is often defined as "the quality of being willing." Does that mean that it's the same as "being willing"? We can say sentences like "Bob's best quality is his willingness to do what's right" which would suggest that they are the same, and phrases like "act of breaking glass" suggest that too. However, if I were to rephrase sentences like "he has the willingness to try new things" to "he has the being willing to try new things", the latter sentence seems wrong.
Willingness is a noun while being willing is more like a verb and is in the progressive tense. If, for example, we use "stand" as a verb and "road" as a noun, we could have the following examples:
Bob stands on the road. [correct] Bob roads on the stand. [not correct]
In other words, willingness (noun) is "the quality of being willing" while being willing (verb) is "having willingness".