Shopping mall (a.k.a mall) and shopping precinct can be the same thing. But by using precinct we are talking about the building, the place where we have plenty of stores, parking lot (usually), etc. In a way we can think of it as "the limits" of a big shopping mall. Precinct seems like more of a British word.
Shopping centre/center can be thought as a commercial center, and also a place where people go to shop (uma galeria, ou o próprio shopping), so it is not necessarily synonym with mall, though sometimes people use the word as such.