The difference between 留心 and 小心
The verbs 留心 and 小心 have a similar meaning, however we should use them in different contexts.
The verb 留心
The verb 留心 
emphasizes that a person gives special attention to something.
emphasizes that a person gives special attention to something.Examples
他想出国留学，所以最近特别留心各种留学方式
He wants to study abroad. That is why he has been learning about various ways of studying abroad lately
tā xiǎng chūguó liúxué, suǒyǐ zuìjìn tèbié liúxīn ge zhǒng liúxué fāngshì
第一次跟他们公司合作，你得留点儿心
This is your first time working with this company. You should be careful
dì yí cì gēn tāmen gōngsī hézuò, nǐ děi liú diǎnr xīn
你就要留心天气预报
You have to pay attention to the weather forecast
nǐ jiùyào liúxīn tiānqì yùbào
留心 can also be used as an adverb meaning “attentively”, “carefully”.
Pay attention
In this case, it is followed by the particle 地 .
.Examples
他留心地听着这些话
He listened carefully to these words
tā liúxīn de tīng zhe zhèxiē huà
老绅士留心地打量了他一番
The old gentleman looked at him attentively
lǎo shēnshì liúxīn de dǎliang le tā yì fān
The verb 小心
The verb 小心 emphasizes that an action is done with caution.
emphasizes that an action is done with caution.Examples
接触化学药品必须小心谨慎
You should be careful when you come into contact with chemicals
jiēchù huàxué yàopǐn bìxū xiǎoxīn jǐnshèn
小心别伤着—这刀子快得很
Be careful not to get hurt. This knife is sharp
xiǎoxīn bié shāng zhe — zhè dāozi kuài de hěn
这个路太陡, 车上去要小心
This road is very steep. You have to drive carefully
zhè ge lù tài dǒu, chē shàngqù yào xiǎoxīn
小心 can also be used as an adverb meaning “carefully”, “cautiously”.
Pay attention
In this case, it is followed by the particle 地.
Examples
我小心地把花瓶放回基座上
I carefully put the vase back on the pedestal
wǒ xiǎoxīn de bǎ huāpíng fànghuí jīzuò shàng
在雨天人们应该小心地过桥
People should carefully cross the bridge on rainy days
zài yǔtiān rénmen yīnggāi xiǎoxīn de guò qiáo
In some cases, 留心 and 小心 can be interchangeable.
Examples
下雪了，路上很滑，一不留心就会摔倒
The road is slippery because it is snowing. If you are not careful, you can fall
xiàxuě le, lùshang hěn huá, yí bù liúxīn jiù huì shuāidǎo
下雪了，路上很滑，一不小心就会摔倒
The road is slippery because it is snowing. If you are not careful, you can fall
xiàxuě le, lùshang hěn huá, yí bù xiǎoxīn jiù huì shuāidǎo