The alternate world essentially inverts the time, so Day becomes Night and Night becomes day. This isn't a version exclusive feature, so if you trade Lunala over to Pokémon Sun, you'll be able to activate the wormhole during the day.
The effects are largely marginal and superficial. The time is inverted so AM becomes PM and PM becomes AM. This will allow for you to do time-based evolutions, hunt for time-specific Pokémon or hunt for time-exclusive Zygarde Cells and events at all times. The areas Altar of the Sunne & Altar of the Moone will become their counterparts, as will the Lake of the Sunne & Lake of the Moone. This essentially makes the overworld of Sun be like the overworld of Moon, and vice-versa
Despite this, this feature does not change the wild Pokémon you'll encounter, so if you're playing Sun you will still encounter Vulpix rather than Sandshrew, and Rockruff & Cosmoem will still evolve into the version specific evolutions.
Another feature is that this is the only way to get the Legendary Pokémon Ultra Beast, Cosmog. Cosmog is not found in the native world, but is in fact found in the Alternate World. When you have gone through the portal go to the Lake of the Sunne or Lake of the Moone from Ula'ula Meadow and reach the altar, with the cover legends still in your party. Cosmog will then appear. You don't need to battle it, it will join you automatically. It can evolve into Cosmoem and then into another Solgaleo in Sun or another Lunala in Moon. You can only get one Cosmog per game.