What does the Twining mechanic do?

The practical difference is control. Without Twining you are a Pathfinder giving orders to a creature that acts for you. In Twine mode you are the creature, moving it yourself in real time, with its abilities on your own inputs. Mobility is the clearest use: whatever a given Aniimo can do to cross ground, you now do personally, so terrain that stopped you on foot becomes a matter of picking the right form.

That reach extends past traversal. Twine mode is built for exploration, puzzle solving and combat as well, which means it is not a movement toy you drop when a fight starts. You merge with the Aniimo and fight in its form. Puzzles that need a specific ability get solved by becoming the thing that has it, rather than by positioning a creature and hoping it does what you wanted.

The limit is your collection. You can only twine with an Aniimo you have already caught, so the set of forms available to you is exactly the set of creatures you own. Every creature in the game is caught out in the world through open-world exploration, with no separate system gating any of them, so expanding what you can turn into is the same activity as expanding your roster. A form you skipped catching is a form you cannot use on the puzzle in front of you.