Does the closure change how you store a bottle?

The one clear practical difference is orientation. Bottles are stored on their side to keep the cork wet, because a dried cork shrinks and lets air past. A screwcap has no such requirement and can be stored upright indefinitely — which, incidentally, is a more efficient use of a cupboard.

The more interesting question is whether the two age differently, and here the evidence supports a real difference without supporting most of the claims made about it.

Closures differ measurably in oxygen transmission rate, and different rates produce different development. That is documented in multi-year bottle trials. What it does not straightforwardly mean is that one is better: lower oxygen ingress protects freshness and also makes reductive characters more likely, which is why some screwcapped wines show a struck-match or rubbery note on opening that a corked bottle of the same wine does not.

Cork also carries a risk screwcap does not, in the form of cork taint. The reported incidence has fallen substantially with improved processing, and the figures quoted vary widely enough that WineHQ does not publish one.

The reasonable summary is that screwcap is not "cork without the downside". It is a different oxygen regime with a different set of trade-offs, and for wine intended to be drunk within a few years it makes very little difference either way.