What is the difference between a professing and a non-professing Christian with regards to their faith, beliefs, etc? I have come across these terms many times but didn’t really come to understand the difference between the two.

Is it that a non professing Christians stopped believing in Christianity or is it that they have some other view from mainstream Christianity?
The word professing means "to declare openly". So a "professing Christian" is one who declares their faith openly. This leads to the conclusion that the opposite of a professing Christian is one who does not announce his or her faith. They may believe and practice their Christianity in private. 

Christianity Today recently divided Christians into five categories: Active, Professing, Liturgical, Private and Cultural. Page 4 of the article gives the characteristics of each type. (Not all would consider every one of those five to be true Christians).

Not all those who profess Christianity necessarily actually adhere to its beliefs. This has led some to adopt the term "professing Christians" for those who outwardly declare their belief while not actually adhering to it. There is some support for this usage in the dictionary definition given above. In that case "professing Christians" are contrasted with "true Christians", meaning those who practice their belief as well as declaring it. True Christians (in this usage) may also actually profess their faith.