Wikipedia does, for what it's worth, treat them as synonymous ("The history of the world (or world history) is the history of humanity", from here).

As a historian I think they are distinct in actual usage, at least in my experience, though it's certainly not a firm distinction. In general "world history" refers to a sub-discipline or a methodology, typically history that takes a more global perspective compared to traditional historiography, emphasising the importance of regions of the world outside of the West and attempting to break free of the orientalist assumptions of traditional historiography. "The history of the world" refers to an object of study -- macro-histories attempting to analyse the entire course of human history.

So a historian specialising in "world history" does not need to literally study the history of the entire world, nor does the history of the world have to be analysed in the terms of world history as an academic discipline.

My sense is that "history of humanity" would be a more anthropological approach, focusing on humanity as a species. "History" in general, of course, is either the overarching academic discipline or the object that that discipline attempts to understand, without any semantic qualifications.