Short Futures and Long Puts are the main hedging strategies for hedging an equity p'folio against a drop, so it is natural that a proposed new technique, going Long a VarSwap, is being compared to the 2 traditional techniques.
How do the 2 traditional techniques differ ? When you hedge an equity p'folio with short futures, you have a constant delta. As S goes down, the amount of protection per drop in S stays constant. When you hedge with a put, the delta increases as S goes down, affording you more protection for the next drop; after S reaches K the protection is 100%. "Convexity" is being used synonimously with Gamma and refers to this "curvature" or second order change in delta.
The question then is does Long VarSwap resemble the Short Futures case or the Long Put case, for the purpose of comparison. The author argues that the "additional protection kicking in as S goes down" (the convexity) is absent in the case of VarSwap.
In my experience when the market is already down a lot the increases in Var become somewhat larger, so I am not sure I completely agree with the author. But it is true that the protection never becomes total, there is no guaranteed floor like with a Put. So the VarSwap is more comparable to the Future, it is an incomplete hedge.