Most people have this confusion. This is very very important for GATE or any such exams - not just for TOC.

"not closed"

"not valid or invalid"

"not decidable or undecidable"

These are complements of closed, valid and decidable respectively. Okay, now all these operate on sets and each of them is a property on the set- i.e., for elements of the set. Even **if one element violate propery**, we say the property does not hold for the set. Now lets see for closed.

A set is closed under an operation if when we operate an element(s) on that operator we get another element from the set.

Now, for set of r.e. languages, even if there are two languages $L_1$ and $L_2$ such that $L_1 - L_2$ as non r.e. we can say that set of r.e. languages are not closed under set difference. But not-closed never says that for any two r.e. languages $L_1$ and $L_2$, $L_1 - L_2$ WILL ALWAYS BE non.re. Rather it MAY OR MAY NOT be r.e.