Is there a difference between 愿意 and 肯 in the following sentence?
"Are you willing to go to China for work?" being translated to one of the two?
In general they're about the same. They are actually used to define each other in some dictionaries. Colloquial usages might differ, but in most cases you can safely use 肯 in place of 願意, especially when spoken.
There is a little difference though. While both words mean agreement/acceptance, with 肯 its acceptance regardless of whether you are happy/willing. In contrast 願意 indicates that its what you want, i.e. there is no unwillingness. So for instance, when exchanging marriage vows, we say 我願意 and not 我肯. The latter would carry an undertone of reluctance.
願意 and 肯 are NOT (exactly) the same.
(I). 願意 is on the more positively-willing to do side. 意 has 心 in it, so 願意 means that the person is willing to do it from one's heart. It generically carries positive feelings.
Ex: 網球選手 R Federer 願意 放棄參加網球賽,因為他的雙胞胎(雙生子)剛出生.
->Tennis player R Federer wishes to give up some tennis games, because his twin sons were born recently.
(II). 肯 may (or may not) imply a decision with reluctance, or under pressure, negotiation, push, etc. 肯 can carry positive feelings in some case, though it can also be (a polite manner way to speak for one's) neutral or negative feelings. It can also be the cases that one does something with bitterness and pain but aims for a delightful outcome or fruits in the end. For example,
Ex: 網球選手 R Nadal 肯 努力拼網球賽,即使他的身體不能負荷.
-> Tennis player R Nadal would like to fight for tennis games, even though his body cannot sustain them.
My answer is very similar to the second part of Semaphore's answer, but I just stress that they are NOT really the same.
p.s. You may also wish to compare them, 願意 and 肯, with are 寧願, 寧肯. This will be another question though.
愿意 and 肯 can be used interchangeably.
愿意 can be explain as a persons‘ willingness，where as a more precise wording for 肯 is asking or expressing consent.
In the Chinese language, words can be interpreted in reversed order to further understand the meaning of words. 愿意 can also be read as 意愿 which means the will of and initial intent of the person.
肯 can also be used as positive conformation as in 肯定.
愿意 and 肯 are similar, but not the same.
"你肯不肯去中国工作" implies that the event "去中国工作" may be unpleasant.
"你愿不愿意去中国工作" implies that the event "去中国工作" is pleasant (or at least not unpleasant).
肯不肯 is usually used to describe unpleasant jobs/ missions/ tasks etc. 愿不愿意 is usually used to describe jobs/ missions/ tasks which are not unpleasant.
肯不肯 is a bit strong and harsh. Try to avoid using 肯不肯.
One of the definitions of 肯 is actually 願意. Basically they mean the exact same thing, but in my opinion, 願意 is more formal than 肯.
Some might find that 願意 is more positive than 願意, but both can actually be used for a positive meaning.
It's rare to see such an interesting question.
愿意, for the simplified Chinese, there is "心" (heart) contained in each character. If you dig the dictionary, 愿意 has more willingness. Which would suggest that in some cases the person already had a positive thought of the issue.
On the other hand, 肯 is more reserved, there would be some trade off (or sacrifice) to make the decision.
In conclusion, 愿意 stands for the favourable side, 肯 stands for the less favourable side.