What becomes unlocked after finishing the main story?

The major things that are unlocked in the post-game are as follows:

Extra areas in side dungeons:

Once you have completed the 'Defeat the monster in the inner sanctum' quests of the 8 major dungeons, you'll get a quest which gives you a key to unlock the doors at the end of these dungeons and you will be able to venture further, as described here.

If you’re an avid explorer there’s a good chance you’ll bump into several of these mysteriously locked doors when completing various dungeons progressing through the game. Sadly however, these remain locked until you’ve completed the entire story – 15 chapters. Once you’ve completed the game you can initiate a special quest that rewards you with the Dungeon Seal Key – the special item needed to unlock the doors inside FF15’s dungeons.

Legendary weapons:

In Lestallum, talking to Randolph will start the questline for legendary weapons. The first quest you will get is A Legend is Born.

More hunts:

In addition to the Adamantoise hunt that you mentioned, there are other hunts, as referenced in this post, which implies that there are a lot more hunts available after completing the main story.

Not the hunts... I think I'm on day 4 of nothing but hunts... so many loading screens....

The Regalia Type-F:

As you've mentioned in the question, you are able to unlock the Regalia Type-F, which allows you to fly instead of drive on the ground. The Regalia Type-F is the only way to access certain areas, such as the Pitioss Ruins.

New Game+:

As mentioned by @DavidYell in the comments, completing the main story will allow you to carry your save into New Game+. Many things carry over from your New Game playthrough. The full details about New Game+ are fleshed out in detail in this Q&A.

As an aside, there are also plans for multiple DLC packs in the future, which will add content for you to do after completing the game (though story completion is not likely to be a requirement).