What's the difference between using the potential form of a verb with よう and ため

Question

For example:

なるように - So that it will become
なれるように - So that it will become
なるために - So that it will become
なれるために - So that it will become

Well, they all mean the same thing

Answer 1

なれる and なる

I think we should first get the difference between なれる and なる figured out. I'm sure you know that なれる is the potential form of なる. Your translation does not capture this very well.

なるように means "so that it will become."
なれるように means "so that it can become."

I think there is a big difference between the two of these phrases. Here are some phrases I pulled from this Chiebukuro post.

幸せになれるように、神様にお願いします。
I prayed that I could become happy.

幸せになるように、神様にお願いします。
I prayed that I would be happy.

There is a big difference between these two sentences. The author of this post notes that the first one means that person prays to god so they can become happy. It implies that they are praying for an environment in which they are capable of becoming happy, but ultimately it is up to themselves to work towards that goal.

The second sentence however, is a straight up wish that god will make them happy. The person who is praying plays no part in that wish.

So, what comes after なれるように might indirectly cause what comes before it while なるように is a direct cause. This is not an absolute rule, but that's what the "can" in "can become" implies.

ために and ように

Note that this is also described in this post. I'll give a shorter version.

The biggest difference between these two is volition (and that ように has a lot more uses). ために expresses that you have control over a situation while ように means you do not. Finally, the agent in the clause before ために and the agent after must be the same. Here's an example:

✕忘れないためにメモしておく。
◯忘れないようにメモしておく。
Leave note so that (I) don't forget.

In this situation, you cannot use ために because you don't control whether you forget something or not. Forgetting is something you don't do on purpose. However, you can leave a note with the intention of making sure you do not forget.

Maybe a more intuitive way to remember this is that ために is similar to "in order to" while ように is closer to "so that".

Conclusion

I'm sorry if this got a little long winded, but the difference between ために and ように is difficult to explain without context. This might be a more suitable translation, but it doesn't really address the nuances.

なるように
なるために
so that it will become.
in order for it to become.

What really matters is what comes before and after these two phrases. You cannot use ために in a situation where the subject of the sentence does not control the situation.

生徒が分かるように説明します。〇
生徒が分かるために説明します。☓
To explain so that the students understand.
To explain in order for the students to understand. (?)

Although I know this example does not use なる, it expresses the same difference between the two phrases. The second sentence is invalid because the subject in the first clause is 「生徒」 and the subject in the second is the speaker. The two subjects are different. What's more, the speaker does not control whether students understand or not.

I hope this helps.