Additional reasons to want to keep the status displayed:

1. You get an EXP bonus when your current class is under 20 and you're grouped with a mentor. This works with level sync, so when a dungeon or FATE temporarily reduces your level under 20, you still get the additional bonus that stacks with all the other EXP bonuses.
2. Players with an aetheryte pendulum (obtained via the referral program) can teleport for free to the aetheryte nearest a friend that is currently using the new adventurer status.

In the event the status was removed, manually or automatically, and you still meet the current conditions for it (less than a week of playtime or not yet reached Stormblood main scenario quests), you can turn it back on with /nas.