Closure property of recursively enumerable language
Basically, "R.e. sets are closed under intersection means that for any two r.e. sets $A \cap B$ is again r.e, but when we say that r.e. are not closed under some set-theoretic operation it means there are at least one pair of r.e sets which results in not-r.e. set under that operation no matter it is set-theoretic difference or complement. In other words, 


R.e. sets are closed under a set theoretic operation $\star$ means that for any pair $A$ and $B$, $A\star B$ is r.e.
R.e. sets are NOT closed under a set theoretic operation $\star$ means that NOT for all pair $A$ and $B$, $A\star B$ is r.e. In simple words, there may be sets $A$ and $B$ such that $A\star B$ is r.e., and there exist other sets $C$ and $D$ such that $C \star D$ is not r.e.


For example, the set $N - 2N$ (the set of all odd positive integers is r.e (even recursive), but the set $N - A_{TM}$ is clearly not r.e. ($A_{TM}$ denotes the set of Halting problem).