The difference between 烦恼 and 懊恼
These adjectives are similar in meaning, but we should use them in different contexts.
The adjective 烦恼
We use the adjective 烦恼 
to emphasize that something annoys or bothers a person, and it makes them feel bad.
to emphasize that something annoys or bothers a person, and it makes them feel bad.Examples
任何一种噪声都会让人们感到烦恼
Any kind of noise can be annoying to people
rènhé yì zhǒng zàoshēng dōu huì ràng rénmen gǎndào fánnǎo
工作的压力使他烦恼起来
The pressure of the work makes him worry
gōngzuò de yālì shǐ tā fánnǎo qǐlái
The adjective 懊恼
We use the adjective 懊恼 to emphasize that a person is upset or disappointed about something.
to emphasize that a person is upset or disappointed about something.Examples
这件事情我没做好，很懊恼
I failed at this job, and I was very upset
zhè jiàn shìqing wǒ méi zuòhǎo, hěn àonǎo
本来能成功的试验却失败了，我觉得很懊恼
I was so frustrated when the experiment, which could have worked, failed
běnlái néng chénggōng de shìyàn què shībài le, wǒ juéde hěn àonǎo