The difference between 应付 and 处理
The verbs 应付 and 处理 can have the same meaning of “to deal with” in some contexts, but also can have different meanings in other contexts.
The same meaning
We use the verbs 应付 
and 处理 “to cope with", "to deal with” to indicate taking action to deal with people or things.
and 处理 “to cope with", "to deal with” to indicate taking action to deal with people or things.Examples
以他现有的经验应付不了目前的工作
With his current experience, he cannot handle the work
yǐ tā xiàn yǒu de jīngyàn yìngfu bùliǎo mùqián de gōngzuò
你可以游刃有余地应付他们
You can handle them easily
nǐ kěyǐ yóurèn yǒuyú de yìngfu tāmen
We usually use 处理 when talking about solving problems.
Examples
我们必须采取积极步骤处理这个问题
We must take positive steps to deal with this issue
wǒmen bìxū cǎiqǔ jījí bùzhòu chǔlǐ zhè ge wèntí
他处理这个问题非常专业
He handled this issue very professionally
tā chǔlǐ zhè ge wèntí fēicháng zhuānyè
Other meanings of 处理
The verb 处理 also has some other meanings.
1) “to dispose of”.
Examples
洗衣机的包装纸箱，既占地方又没有什么用，快处理了吧
The washing machine packaging carton is useless and takes up space. Let’s dispose of it
xǐyījī de bāozhuāng zhǐxiāng, jì zhàn dìfāng yòu méi yǒu shénme yòng, kuài chǔlǐ le ba
炸弹已安全处理
The bomb has been disposed of safely
zhàdàn yǐ ānquán chǔlǐ
2) “sell at reduced prices”.
Examples
这批过季的衣服尽快减价处理吧
These out-of-season clothes should be sold at a reduced price as soon as possible
zhè pī guòjì de yīfu jǐnkuài jiǎnjià chǔlǐ ba
他还没有处理掉积压的商品
He hasn't gotten rid of the stale goods
tā hái méiyǒu chǔlǐdiào jīyā de shāngpǐn
The other meaning of 应付
The verb 应付 can be used in the meaning of “to do something perfunctorily”.
Examples
你根本是在应付，现在该努力干了
You're simply performing you duties without much thought, but now is the time to work hard
ni gēnběn shì zài yìngfu, xiànzài gāi nǔlì gàn le