The distinction is clearer than you might think. First of all, only a small group of American journalists, the US Department of State and the President of the United States use the term ISIL with any regularity. Both phrases claim to be translations of the same Arabic phrase, but the interesting part is that the term “Levant” which includes Syria and Palestine is actually an outdated term. The modern term for the area is “Greater Syria.” The intent in both cases is to negate any Jewish claim on a Palestinian homeland, but in that region, only those wishing to restore the ancient Caliphate stretching from Iraq to the Mediterranean Sea use the term Levant, and they couple it with the term Mesopotamia for Iraq. Most of the Arabic world who recognizes the modern national boundaries of Iraq, Turkey, Iran, and Syria have forsaken the Mesopotamia reference as well as the Levant terminology.
The radical Islamic terrorists seeking to restore the Caliphate use the term Levant primarily to reinforce their claim over all of Palestine, rather than merely the territory they now control. By adopting the terminology, journalists and politicians give credence to that claim.
It appears to many Americans that since the use of the term is so consistent within the Obama administration, that it is not accidental but deliberate. If there is a plausible explanation as to why the President of the United States almost daily reinforces the claim of terrorists over the Jewish homeland, we are yet to hear it.