I read that asymmetric relations are closed under intersection and set difference but not under complement.
Although I could understand why is it like that, but we can represent set difference as
A - B = A intersection B' .
Now it is closed under difference and intersection. So it should also be closed under complement according to the equation. Why is it not like that ?
Asymmetric relations are actually closed under complement. Let $R$ be an asymmetric relation, i.e. there exists a pair $(x,y)\in R$ with $(y,x)\not\in R$.
If $R'$ is the complement, then $(y,x)\in R$, but $(x,y)\not\in R$. Therefore $R'$ is also asymmetric.
Assuming my definitions of your terms are correct, this violates the statement of your question.