通り literally means "street" or way in which something goes or is transmitted. However, it is often used similar to English "as...say".
The 通り of this discussion is an adverbial noun and describes the way in which something is supposed to be like. It may seem very similar to ～ように, but there are few slight differences. It is essentially a means of paraphrasing to show that something is being done 100% the same way as something.
～ように, on the other hand, is used in telling/showing that something is being done similarly to what it attaches to. There is also an added sense of hope from the speaker, which makes that 100% unattainable even more so. There is interchangeability when one is illustrating and showing hope, but if it is stilted to one side, it becomes inappropriate.
When attached directly to nouns, とおり becomes どおり. To attach ～ように after a noun, you need to make it ～のように. Also, both of these patterns can be in between noun phrases as ～どおりの and ～のような respectively.
Do it just as I do.
She arranged the flowers just like her teacher did.
I did it just as you said.
It's just as I imagined it.
It's completely like you said, isn't it?
I will write it down so I won't forget.
I had some one lift me up high in order to see well.
Just as I informed you
To have it one's own way.
These are the results of the prosecution; it was just as expected.
Things are basically just as normal.
As you know
In prayer or wishing, ～ように is used after the polite auxiliary ～ます.
I pray you succeed.
I do hope I will pass the college exam.