这两个短语都是一种结构：特殊疑问词+不定式构成的，相当于宾语从句。
what to do是完整的（what是do的宾语），表示做什么，故不定式后不可再加宾语。如：
I don't know what to do.
(=I don't know what I can do.)
how to do是不完整的（how提问方式），必须加上do的宾语，表示如何做。如：
I don't know how to do this thing.
(= I don't know how I can do this thing.)
一般情况do后必须加上宾语。
注意：how to do it
这里do 后面跟了宾语it，便是一个完整而又正确的表达。
意思是“怎么做（它），如何做（它）”，询问的是方式，例如完成某事的方法。
例句：Could you please tell me how to do it？（你能告诉我怎么做它吗？）
how to do it询问的是方法，是完成某事应该怎么做。
Mom, can you tell me how to do it?
妈妈，能告诉我怎么做吗？
《柯林斯高阶英汉双解学习词典》
Could you suggest someone to advise me how to do this?
这该怎么办？你能不能推荐一个人帮我出个主意？
《柯林斯高阶英汉双解学习词典》
If you don't know how to do it, don't fool with it.
你要是不会干就别胡来。
《汉英大词典》
No matter how hard it is, I'll learn how to do it.
什么难学不难学，我保证学会。
《汉英大词典》
Let's pool our ideas and see what to do.; Let's put our heads together and see how to do it.
大家来想办法。
《汉英大词典》
It wasn't until we each went our separate ways that I began to learn how to do things for myself
直到大家各奔东西，我才开始学着自力更生。
《柯林斯高阶英汉双解学习词典》
I don't know what to do next
我不知道接着该干什么了。
I'm just fed up and I don't know what to do.
我就是腻味了，不知道该怎么办。
We know what to do to make it work.
我们知道如何让它起作用。
We had to think what to do next.
我们必须考虑下一步该怎么办。
The government appears to be in a quandary about what to do with so many people.
政府似乎陷入了左右为难的境地，不知道该拿这么多人怎么办。
There are seasoned professionals who can teach you and show you what to do
有经验丰富的专家教你，并向你演示怎么做。
We will be having a meeting to decide what to do
我们准备开个会来决定怎么办。
But you'd find him a good worker if you showed him what to do.
但是只要教给他怎么做，就会发现他是个不错的员工。
You're a wise old man: tell me what to do.
您是位睿智的长者：告诉我该怎么办。
Look, Mister, we know our job, so don't try to tell us what to do.
嘿，先生，我们知道自己该干什么，不用你指挥。
This was such a great breach of etiquette, he hardly knew what to do.
这是严重失礼的行为，他几乎手足无措。
He knows what to do. He's been around
他知道该做什么。他阅历很丰富。
'Tell me what to do!' she implored him.
她哀求他说：“求你告诉我该怎么办！”
I'm glad you read the riot act to Billy. He's still a kid and still needs to be told what to do.
我很高兴你警告比利不许胡闹。他还是个孩子，仍然需要有人告诉他该做什么。
The girl was uncertain what to do, or what tone of voice to adopt.
女孩不知道该做什么，也不知道该用什么口气说。
Was there a difference of opinion over what to do with the Nobel Prizemoney?
在如何利用那笔诺贝尔奖金的问题上有不同意见吗？
He is tussling with the problem of what to do about inflation.
他正全力应对如何解决通货膨胀的问题。
以上例句均来自《柯林斯高阶英汉双解学习词典》
James was at a loss as to what to do and felt lost.
詹姆斯无所适从，惘然若失。
She spread out her arms, not knowing what to do.
她扎煞着手，不知该干什么。
People concerned have been invited here to discuss what to do next.
邀集有关人士讨论一下下一步干什么。
For a moment I just didn't know what to do.
我一时心里没主。
以上4例来自《汉英大词典》
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