I doubt there's much in it one way or the other, and even if there is, I bet it varies by implementation (IE vs. Chrome vs. Firefox vs. ...). Both will cause reflow events in the tree.
Showing and hiding is simple and straightforward. Adding and removing can have its uses, but for the most part is probably overkill. For one thing, you have to keep a reference to it so you can add it back later, and you have to remember where you need to add it back.
But (and this is a bit off-topic) your mechanism for showing and hiding has some issues:
With your example code, things can only be inline, not inline-block or block, which would be somewhat...limiting.
Your code also completely replaces the style information on the element (and may not work on IE; see David's comment on your question), so all other styles directly applied to the element will get blown away; consider using the style property of the element instead: childRef.style.display = 'none"; and childRef.style.display = "inline"; (or block, or...). That way, you don't bludgeon any other styles on the element.
Completely removing a child from the tree (while keeping a reference to it so you can put it back later) definitely has its uses, though. For instance, when it's not in the tree, it won't be found when you're walking the tree with selectors for doing this, that, and the other; which may be helpful.