Is wine with a screw cap always cheaper/worse than wine with a cork?
No. Or rather, sometimes, but not necessarily.
Today, a cork in a bottle is basically synonymous with wine, but corks used to seal everything from milk jugs to medicine bottles, and not very well. Cork is porous, meaning it allows a little bit of air to flow in and out of any bottle it seals. It also expands and contracts with temperature and humidity changes, and can dry out and crack/completely crumble given enough time. This is (part of) the reason why wine cellars are kept under such carefully controlled conditions, and also why wine bottles are stored on their sides (when there’s liquid touching one side of the cork, it keeps it from drying out).
Exposure to oxygen is the enemy of fine wine. When sommeliers say a wine is “turned” or “shot”, they mean it’s been exposed to too much air and is now oxidized. You know when you leave like, a half a glass of wine in the bottle and then try to drink it a few days later, but it tastes kind of like vinegar and raisins? That’s oxidation.
Anyway, a little bit of breathability can be good for wine. Under carefully controlled circumstances (like in a cellar), airflow through the cork can help a wine gain age and complexity. But something like 95% of the wine produced in the world is drank within a year of when it was sold. Most wine doesn’t need to be aged, and shouldn’t be kept more than a couple years before drinking anyway. And that’s where the screw cap comes in. Unless you’re planning to age a bottle of wine for a decade or so, there’s absolutely no reason why you need a cork. Screw caps, or “Stelvin Closures” if you wanna get froofy about it, are cheaper to produce, seal bottles better, and are obviously easier to open (so stop apologizing to your waiter for ordering them).
Screw cap wines do tend to be less expensive than wines with corks, but that’s not necessarily an indicator of quality. The wines that actually need corks, the stuff you collect and keep in a cellar, are the highest caliber, most expensive wines on earth. For other wines, most of the time the cork is just for show.
Nowadays there are all sorts of alternatives to corks. Look closely and you’ll find a lot of corks aren’t even cork anymore; they’re rubber. In parts of Austria and Germany, glass stoppers have been traditionally used in place of cork since forever ago. And aside from the screw cap, there’s now very good quality wine being sold in boxes, cans, and tapped right from kegs behind the bar. Wine professionals know this. So stop worrying about being judged, ditch the fancy wine opener and drink what you like.