Meanwhile, to me it sounds perfectly fine. It’s to proactively fail to do something. You were aware you had to do something (e.g. show up at court), and you just decided not to. I admit, “willfully” sounds a bit better to me, but the OP says he doesn’t see a difference, so, either way, it means you failed to do something you were supposed to do. To fail to do something, as stated, does not necessarily require an attempt: it could simply be not doing something, and that usage of “fail” is perfectly ordinary to me.
‘Willfully fail’ sounds fine to me, but some people have claimed there is a difference in meaning.
To me, “willfully” is stronger than “willingly.” I’m just spit-balling here, but as I hear it, “willfully fail” has more of a purposeful failure, and “willingly fail” encompasses a more apathatetic “I don’t give a shit” failure. Like you failed but you didn’t really try, as opposed to setting out with the mindset of failing. That’s just how it feels to me, quickly thinking about it. My thoughts may change with other examples and other people’s input.
What if you really intended to succeed but then you got distracted posting on the SDMB and didn’t get round to it? (A totally hypothetical possibility that I’m sure has never happened in reality…)
The latter I agree is willingly failing…not showing up for the exam with no extraneous circumstances making it impossible or terribly impractical. But if you half-ass study for the exam, you still didn’t desire to fail, so I just have a hard time wrapping my head around the concept that you willingly failed by not studying very hard.
I could see the argument that as you KNOW that you will fail unless you diligently study, that your intentional act of half-ass studying meant that you would fail, and as you intentionally didn’t study, you intentionally (willingly) failed.
But that construction crosses the line into negligence. It’s not guaranteed that you will fail by half-ass studying; maybe I pass with a C-, when proper studying would get me an A?
I know that if I drive drunk then there is chance that I might kill someone so if I do then I willfully killed someone–which nobody would say. There is also a chance that my drunken driving doesn’t cause any accident at all or “merely” injures someone.
I guess I need to call it a day on the reading. 
ETA: And I see that the prior poster said not studying at all, so change all instances of “half-ass studying” to relying on memories from class.
That would most likely not be “willfully” nor “willingly” failing at a task to me.
I think the task-master might disagree. 
The perspective would certainly color the characterization of the failure, yes.
Well, 40 years ago in high school, my graduating marks in maths and sciences were not going to be high enough to get me into the university I wanted.
The most effective way for me to resolve the issue was to go into the various exams and deliberately fail. This would, because of the regulations at the time, allow me to repeat those courses in a different school board (I did not want to spend another year at the same school for various reasons) for a very low, nominal administrative fee.
IMHO I willingly failed those courses
If you say “He failed to wash his hands before he ate.”, certainly you don’t mean he attempted to wash his hands and failed to do so, but that he simply did not. Using “willingly” or “willfully” here before “failed” implies that he knew he should have washed his hands before he ate, but affirmatively decided to not wash them anyway. And that’s the kind of willful failure, if for something a bit more serious, that causes legal problems.
If you say “He studied as hard as he could but still failed to pass the exam”, then it makes no sense to add “Willfully” or “willingly”, since it’s exactly opposite to the semantic meaning of the sentence. When all you think about is that kind of sentence, even divorced from the context where the meaning implies the failure was not intentional, then certainly it might seem a bit weird to think about how one might willfully fail. But with the kind of sentence in the first paragraph, it should be obvious.
Based on OP it sounds like someone made an effort to not do something difficult and shameful. I could see willingly failing at something which is dangerous, shameful or difficult.
I don’t know if willingly failed is accurate in all situations. The person may want to fix their substance abuse issues but not be able to. I’d consider that a bit different than a sex offender who knows he has to register but finds it shameful so he doesn’t do it, he doesn’t want to open himself up to social ostracization.
You were required to do something but didn’t, nor did you alert the court that circumstances beyond your control prevented you from doing it.
To willingly fail to cut the grass I think you’d need to get the mower out, fire it up but set the blade height above the grass length and go round the lawn mowing it at a height that none is actually cut.
Anything else where you decide not to not to mow the lawn is willful.
Jasmine’s Dictionary
He “willingly failed” to complete his assigned task.
He “blew off” his assigned task.
So, yes, people “willingly fail” to do things all the time.
My sense, which could be wrong, is that “willfully failing” implies a more deliberate act of noncompliance where “willingly fail” means to neglect to take the steps necessary to comply.