I assumed that the difference was that a potion can't be used during battle (because it says that) but an elixir can (because it doesn't say it can't). However, when I try to use an elixir during battle I get an error message saying I can't use inventory items during battle. And I can't see any way to move the elixir from my inventory into somewhere else where I could use it during battle, so it seems like I can't use that during battle either. So if neither of them can be used during battle, what's the difference between them?

Elixirs can be used in battle, just not from the inventory page.

In Astryl's Vale, while in a battle, hit "Manage". There'll be an elixir icon (there's a potion icon right next to it for some reason, but it's always grayed out). If you click that, you can choose an elixir and a pet.

In regions 1 & 2, there's a red arrow button under the bottom right corner of the explore picture. From there, you choose a party pet to use an elixir on, then choose the elixir you want to use.

Thanks so much!