Adventure is also known as PVE, which stands for Player vs Environment. In this mode you are going to play against NPC.
This means you are battling against other players, and you will be ranked in the leaderboard based on your Match-Making Rating (MMR), both on- and offseason. Ranking high in Arena during Seasons come with the benefit of getting rewards in AXS.
You can play both Adventure and Arena mode with or without energy, however what you'll earn by winning a match will differ.
https://www.axiehandbook.com/playing-the-game/battle-basics
https://axie.substack.com/p/axie-infinity-community-alpha-guide