The former is Sino-Vietnamese are seen as more formal while the latter is native Vietnamese and used more-so in everyday speech. Additionally hiện tại is used to mean the present.
"We are not currently watching TV" is not accepted. Does "Hiện tại ... khong xem ... " imply we were watching before but have now stopped?
"Hiện tại" means "right now, at this very point in time". "Hiện tại....không xem...." means "at this very point in time we are not watching......"
Which is exactly what he said. I think currently should be accepted here too.