Approval is a noun. Think of it like a physical object that you have to obtain by asking someone for it. If you needed Bob to give you a book, you would say:
I need to get the book from Bob.
Just the same for approval, for all that it is not generally a physical object:
I need to get approval from Bob.
Approved is the past tense/past participle of the verb to approve, and you want to say who is doing something. That uses the preposition by to indicate the actor:
I need to get it approved by Bob.
The essence of the difference is that one is a verb, and the other is a noun. This means that get is a different sense in each case - one is in the sense of to obtain, and the other is the sense of to cause.