Most of the time, these two words can be used interchangeably (可交换的). The meanings are quite similar. However, there are situations where one is better.

Begin, means start the first part of an activity or action. for example, “at the beginning of this project…”. It is used where a process with a natural (天然的) end.

Begin 开始一个活动或动作，强调的是从无到有，并且活动或动作有始有终，有明确的结束的时刻。

Examples:

Let’s begin!

It’s time to begin your speech.

My son is eager to begin reading the book. (it means he is going to read chapter 1)

我儿子迫不及待地想开始读书(Begin强调的是他从第一章开始读)。

Start, means cause something (event) to happen. It is also used to talk about creating a new business.

When talking about machines, we always use "Start". You can’t “Begin” your car.

Start 开始干某事，强调的是让某事发生，也许这是第一次开始发生，也许之前已经发生多次了。启动机器时，我们只能使用Start。

Examples:

My car won’t start again.

Press this button to start the printer.

She started a new restaurant. 她新开了一个饭店。

My son is eager to start reading the book. (it means that he had been reading it many days and he will continue reading today)

我儿子迫不及待地想开始读书(他可能读这本书有段时间了，可能已经读完10章了，今天想继续读而已)