What is version to do that version one does not?

I am going to assume you are referring to the DC/PC version and not any of the other consoles. This should give you a run down on the version 2 changes and additions.

The game's level cap is increased from 100 to 200. As a result, many characters had their stats rebalanced.
A new difficulty mode is added for characters level 80 and above called "Ultimate". In this difficulty mode, the enemies spawned in each area are replaced with new enemies and each area's appearance is changed.
The maximum level for Techniques had been increased to Level 30. This level cap is only available to Force characters however as Hunters and Rangers still have Level 15 as their Technique Level Cap.
Ver. 2 introduces various new Weapons, Armor, Shields and Mags.

A shortcut window can be used to access items quickly.
A shortcut menu can be accessed in Ver. 2 during gameplay by holding R and pressing Y to equip or use items. Pressing R with the menu up will change to different item types.
Android characters can now set traps similar to the ones that can be encountered ingame.
New game modes were added to the game's Online Mode called Challenge Mode and Battle Mode. Challenge Mode is a mode that emphasizes puzzle solving and team work with other players as they try to complete specially made versions of each stage as quickly as possible. Battle Mode is the game's "PVP" mode. Players can either traverse each area as normal but with the risk of friendly fire or play through preset "Battle Rules.

The Go-Go Ball minigame present in Lobbies 11-15.
Accessible online by Ver. 2 players are Lobbies 11-15. These lobbies contain a soccer minigame called "Go-Go Ball".
Players can add a "Whistle" sound effect to their Symbol Chat artwork. In addition, a new "Capture" command had been implemented which lets players save other player's Symbol Art.
Players can search for other players in their current block from the Guild Card menu using various parameters such as the player type and level.
The option to search the ingame location of a Simple Mail's sender has been added.
Ver. 2 lets players customize their character's appearance later on by accessing the "Dressing Room" from the character selection screen. This option becomes available for 100,000 Meseta.
Unlike its predecessor, Ver. 2 supports the Broadband Adapter.