Hull Rupture ships with four difficulty levels. What each tier changes is not published — the names of the four levels and their per-level modifiers have not been announced — but two things are confirmed and they matter a lot: the open playtest contained only a single 'base' difficulty, and on higher levels the galaxy map starts fighting back.
What the four Hull Rupture difficulty levels change
The store page and the launch post both state that the game mixes four difficulty levels. The open playtest was not one of them: it ran a single 'base' difficulty, and every difficulty discussion before the launch build was really a discussion of that one tier.
The developers said in Devlog #3 that they would share more about the difficulty levels, the three starting ships and the game modes in a future devlog. That devlog had not been published when the launch post went out, which is why the four tiers are still unnamed.
What higher difficulty adds to the map
On higher difficulty levels, some stars you visit trigger combat encounters. The aliens throw extra waves at you while your economy keeps running, so the map stops being a pure resource trip.
The bigger change is guarded locations, which never appeared in the demo and are planned for the full release. They only appear at higher difficulty levels, and arriving at one means extra enemy waves that do not count toward the 20 you normally need to survive. Any reward there comes with extra danger and extra strain on the ship.
That single rule is the reason higher difficulty is more than a damage multiplier: your twenty waves are no longer the only twenty waves you have to handle. See the galaxy map guide for where those fights happen, and the survival plan for how to keep the economy standing while you take them.
Picking a tier for your first run
The difference that matters most is how much you have to manage at once. On a higher tier you are running the standard 20 waves and fighting map encounters in between them, so an economy that only just handles the standard 20 will also be handling map fights on top of it — with a ship that is heavier and a crew that is already committed elsewhere.
Start on the lowest tier you can find. Learn the production chain and the turret placement rules on a tier where the only thing between you and a lost run is a bad economy, then move up. The galaxy map guide lists the star types that trigger the extra fights, so you can see where those map encounters come from before you pick a tier.