The distinction is not between professing christians and non-professing Christians, but between professing Christians and non Christians. The idea of professing is using your mouth to say words that identify you as a Christian.
The term professing Christian is often used in contexts where the actual status of a persons faith or beliefs are in doubt. A person might claim to be a Christian in name, but if it is not a genuine confession of the heart, their life might not show any evidence of their faith being lived out. When somebodies life does not show any evidence that they actually believe what they profess, this label might apply.
Bob is a professing Christian, but you'd never know it from his behavior at work.
Confusingly, it can also have a different more positive connotation. This usage is more rare and you will have to listen to contextual clues in the conversation to know if it is used this way. Some people will use this term as a way to affirm something positive about a person's beliefs, specifically that they are open and honest about them.
Bob is such a great neighbor. He's a professing Christian and has such a caring family.
The alternative is that somebody who does not profess even to be a Christian probably does profess to be something else. Whatever they claim to be, that is their profession. A person could be a "professing Buddhist" or "professing Atheist" as well.