score:5
It depends on precisely where and when you're going, but in general, yes. If you're going to London Bridge (a station near the historical financial district, the City of London) on contactless or Oyster you'll be paying £8.30 off-peak or £15.10 for peak time (starting your journey 0630 to 0930 on weekdays). Meanwhile for a paper ticket you would pay £9.70 for a weekends-only super off-peak day single, or £11.90 for an anytime day single. So if you're travelling in the peak time this would actually be cheaper than contactless, but otherwise it would be cheaper to pay with contactless. A period Anytime Return (what Americans would call "round trip") ticket costs £21.10. All of these options are valid on both Thameslink and Southern trains to London Bridge.
B.Liu points out that most of the above prices are also the same for a Thameslink train that goes beyond London Bridge through the Thameslink tunnels to Blackfriars, City Thameslink (both serving parts of the City of London), Farringdon, or St Pancras International (serving the King's Cross area); the only difference is with the Anytime Day Single which would be slightly more expensive at £12.10.
Meanwhile if you're going to London Victoria near the West End (one of the main leisure districts), the prices are the same for the ordinary Southern trains, or £19.80 at any time for a Gatwick Express train (despite the name, depending on precisely which time you arrive it's sometimes faster to catch a Southern train than a Gatwick Express one!). Meanwhile an Anytime Day Single not valid on Gatwick Express paper ticket would cost you £16.70 and an Anytime Single valid on Gatwick Express £19.90. An Anytime Return (what Americans would call "round trip") not valid on Gatwick Express costs £33.40, and one valid on Gatwick Express (and also on Southern) costs £37.80 which is less than twice £19.80.
So in general it's cheaper to buy a paper ticket if:
In all other cases, you should use Oyster or Contactless.
This is all assuming you want to go standard class, and that you don't have a railcard or any existing tickets valid for part of the route.
Sorry your first experience of our fares system is with one of the worst examples of its confusion!