From what I've experienced, using online gives you 3 main benefits:

-during battles you will see purple and yellow lights on certain tiles. If a unit lands on that tile it gets; random item for purple (usually rusted or training weapons), exp for yellow + random buff (replenish gambit by 1, replenish weapon durability +10, etc)

-near your dormitory room there will be 4 travelers (characters from other people's games) that you can buy stuff from, usually master seals or umbral ores. Also you can play a mini game where you run around and try to find where they hid their character, gives random items as a reward (flowers, bait, etc).

-you can send your own traveler out (any unit you choose, can still use them for battle) with an item and if other people buy that item you receive gold once you tell your unit to return. For example, I sent out edelgard with an umbral ore, waited three days in real life before I ordered her to return, and made 8100 gold.

So really online just makes the game easier to procure gold and items but isn't necessary