Could someone please explain this to me 
I don’t understand the difference, you’re both trying to see how objects can be used to reveal more about humans at the time right?
I think material culture is a much more recent term-- sort of like the difference between “art history” and “visual culture”. A bit more inclusive-- accounts for more recent objects and close-at-hand cultures and with less of a sense of hierarchy or canon (“Archaeology” has that “archaeo-” in the word, and an olde school set of baggage to carry around). Someone in an Anthro or archaeo dept will have to tell us, but is there currently any kind of political hoo-haw about changing department names along these lines and all that (I’m extrapolating from currents in art history)?
Material culture is the stuff – the acumulated “things” that create culture. Archaeology is the area of study that looks at material cultures from “old” societies and extrapolates about the lifeways that created those cultures. That’s why archaeology is a sub-area within anthropology, because it theorizes about human society.
You can study Material Culture without studying archaeology (for example, if you looked at modern societies), but the reverse is not true.
As for current trends at universities, I have no idea… its been 10 years since I got my degree in Anthropology-with-an-archaology-focus and my undergraduate university is rather notorious for doing things the “old school way” (they still give diplomas in Latin and have a “government” dept, not “poli sci.”)