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As I understand it, they're synonyms except ～にしたがって is more formal, but in
Shin Kanzen Master Bunpô N2 it says:

> 「～につれて」は一方方向の変化の場合にだけ使う。「～にしたがって」は一方方向の変化でなくても使える。

The thing is that the examples that appear in the book with ～にしたがって fit with
what I understand as an 一方方向の変化, so I don't know if they just didn't choose
the best examples to show the difference, or if I'm not understanding the
concept of 一方方向の変化 correctly. The two examples with ～にしたがって are:

> 車のスピードが上がるにしたがって、事故の危険性も高くなる。

And:

> 息子は成長するにしたがって、口数が少なくなった。

Can someone explain me the difference between those two structures, or show me
an example in which only ～にしたがって would be correct and why?
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question
grammar word-choice
Difference between ～につれて and ～にしたがって
It's well explained
[here](https://nihongonosato.com/jlpt/n2-grammar/n2-totomoni/). I don't think
there's a difference between the two as long as time evolution is concerned,
meaning in your examples, you can use both 〜につれて and 〜にしたがって. An important
difference is 〜にしたがって can also mean "based on" or "according to" (as defined
in 意味② on the page above), e.g. ルールにしたがって (= following the rules). If this is
not what they meant with "一方方向の変化でなくても使える", I don't know what they wanted to
say.
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answer
The difference is not large, but there are cases where one sounds more natural
than the other.

  * `XにつれてY` does not strongly assume cause-effect relationships. It can be used when X and Y change almost independently over time. `Xに従ってY` is more explicit on the cause-effect relationship, and thus preferred in technical contexts.
  * `XにつれてY` is usually used with temporal changes. (Time is irreversible, so maybe this is what 一方向の変化 in your textbook is referring to?) On the other hand, `Xに従ってY` has a wider range of usage.

For example...

> 夜が近づくにつれて、ますます雨が強くなってきた。  
>  As night approached, the rain became heavier and heavier.

In this sentence, つれて is more natural because there is usually no direct
cause-effect relationship between nightfall and rainfall. But if there is an
approaching typhoon mentioned in the context, 夜が近づくに従って would sound fine, too.

> 統計によると、英語を学び始める年齢が若くなるに従って発音も正確になる。  
>  Statistics show that the younger one starts learning English, the more
> accurate his/her pronunciation becomes.

In this sentence 従って is more natural because this is an academic-sounding
sentence and "starting age" itself is not something that changes over time.

> 息子は成長するにしたがって、口数が少なくなった。

In this sentence ～につれて and ～にしたがって are mostly interchangeable, but したがって
sounds like the speaker is aware of the fact that boys tend to become more
silent as they grow up.
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answer