Expressing "to discover" with 发现 and 发觉
These verbs have the same translation, but they are used differently.
The verb 发觉
The verb 发觉 
expresses that a person has either understood something or began to take something to heart.
expresses that a person has either understood something or began to take something to heart.Examples
他越来越关心我了，我发觉他喜欢我了
He took such good care of me that I realized that he likes me
tā yuè lái yuè guānxīn wǒ le, wǒ fājué tā xǐhuan wǒ le
她发觉那工作越来越没意思
She finds her job more and more boring by the day
tā fājué nà gōngzuò yuè lái yuè méi yìsi
都怪我，过了这么长时间才发觉事情不正常
It is all my fault. I needed so much time to figure out that something was wrong
dōu guài wǒ, guò le zhème cháng shíjiān cái fājué shìqing bú zhèngcháng
The verb 发现
The verb 发现 expresses a situation where a person has found out something that he didn’t know before.
expresses a situation where a person has found out something that he didn’t know before. Pay attention
The verb 发现 is often used to talk about research or discoveries.
Examples
中国是最早发现和利用茶叶的国家
China is the first country to discover and start using tea leaves
zhōngguó shì zuì zǎo fāxiàn hé lìyòng cháyè de guójiā
去年这儿又发现了一个大油田
Last year another large oil field was discovered here
qùnián zhèr yòu fāxiàn le yí ge dà yóutián
调查了这么长时间，发现了什么？
What have they discovered after so much research?
diàochá le zhème cháng shíjiān, fāxiàn le shénme
发现 as a noun
发现 can function as a noun “a discovery”.
Pay attention
发觉 cannot function as a noun.
Examples
科学家很快就会有新的重大发现
Scientists will make new major discoveries soon
kēxuéjiā hěn kuài jiù huì yǒu xīn de zhòngdà fāxiàn
这就是诗的奇妙之处，你总有新的发现
The miracle of poetry is that there is always something new to discover
zhè jiù shì shī de qímiào zhī chù, nǐ zǒng yǒu xīn de fāxiàn
The same meaning
In the meaning of “to notice” and “to figure out”, 发现 and 发觉 are used the same.
Examples
她走了半天，才发现方向错了
After walking a long distance, she figured out that she was going the wrong way
tā zǒu le bàntiān, cái fāxiàn fāngxiàng cuò le
她走了半天，才发觉方向错了
After walking a long distance, she figured out that she was going the wrong way
tā zǒu le bàntiān, cái fājué fāngxiàng cuò le
Examples
我们发现最近小张的行为有些不正常
We have noticed that Xiao Zhan has been acting weird lately
wǒmen fāxiàn zuìjìn xiǎozhāng de xíngwéi yǒuxiē bú zhèngcháng
我们发觉最近小张的行为有些不正常
We have noticed that Xiao Zhan has been acting weird lately
wǒmen fājué zuìjìn xiǎozhāng de xíngwéi yǒuxiē bú zhèngcháng