Correct

办事 vs 做事

/ bànshì vs zuòshì /
Strokes

Definition

办事 and 做事, both are Verb-object phrases, which means [handle affairs; work; do a deed].

Example

办事 is applied to an official business.
1
dào
běijīng
北京
shì
lái
bànshì
办事
de
  
,
máng
de
hěn
  

He came to Beijing on a business and had a lot of things to do.

2
bànshì
办事
gōngdào
公道
  
,
jiā
大家
dōu
xìndeguò
信得过
  

He is fair and square in what he does.We all have confidence in him.We all trust him.

3
bànshì
办事
néng
能力
hěn
jiàng
  

He is competent at administrative affairs.

Some elements can be inserted between 办 and 事 or 办 and 事 can even be reversed to indicate the business to be important or unimportant.
做事 cannot be used in the examples.
4
Děng
bàn
wán
leshì
了事
  
,
men
我们
可以
zài
zhège
这个
chéngshì
城市
yóulǎn
游览
liǎng
tiān
  

After handling all the affairs, we can enjoy ourselves for several days in the city.

5
Zhè
jiàn
shì
bàn
de
cuò
不错
  

He did a good job of it.

做事 can mean "to work in an organization or company".
Usually, 做事 can be separated to mean "to do something concrete".
6
zài
jiā
gōng
公司
zuòshì
做事
  

He worked for a company.

7
Jīntiān
今天
zuò
le
shǎo
不少
shì
  
:
fēnpèi
分配
de
gōngzuò
工作
dōu
wánchéng
完成
le
  
,
hái
xiě
le
fēng
xìn
  

I did a lot of things today: I not only finished my assignments, but also wrote several letters.

8
xiǎng
qǐng
bāng
zuò
diǎnr  
点儿
shì
  

I'd like you to do me a favor.