How did you get 144?
I can come up with 576 = (4!)*(4!):
Women will occupy the northern half of the round table, where they can be lined up in 4! ways. Also, men will occupy the southern half of the table, where they can be lined up in 4! ways. In all this gives (4!)*(4!) = 576 ways to seat the women and men around the table.
This is the normal interpretation of the problem in terms of circular permutations, where a seating is considered the same as the seating obtained by shifting everybody one seat to the right. If these two
seatings were considered different, the number of seatings would be multiplied by eight. perhaps there is a third interpretation of the problem which gives an answer of 144, but I don't see it. Explaining your process of counting also shows your interpretation of the question. For learning purposes it is much more important than the numerical answer.