What is it?
The Persuasion/Influence Differentiation is simply the recognition that persuasion and influence are different, and understanding how this difference effects efforts to gain the agreement and compliance of others.
Why is this important?
Without understanding the difference between persuasion and influence, and the effects they have on your efforts to gain the agreement and compliance of others - you will not likely recognize what changes are appropriate to develop effective skill in either one.
Influence is the product of how much others trust us, believe we care about them as a person, and believe we have, work to and live to equal or higher standards than their own. This is a function of disarming the Guarded Mind and influence takes time to develop. The development of trust and high character are discussed elsewhere on this site, and nearly all traditional virtues will build our influence with others. When we have influence, it is like having a "key" to the other person's mind so to speak - their mind is mostly open to what we say.
Persuasion is the use of language to lead others to agree or comply with us during our conversation when we may or may not have influence with them. When we do not have influence with someone, getting their agreement and compliance is much like trying to unlock their mind without a "key" - we must use the tools and skills of lock picking so to speak. The primary picking tools are questions - that help them discover new understanding (different from what they initially believed) for their self.
Persuasive skills that are effective when we do not have influence are somewhat rare - and no one has them naturally any more than anyone is a natural lock picker.