Normal quests are based on the main story of the Bleach anime series, with each chapter divided into chapters which are divided again into sub-chapters. Each quest has a level requirement to join with rewards including experience points, rare gems, character-specific ultra-hit gems, etc. They are only available once unless you refresh the game by re-downloading it or using life tickets to reset your progress. You can also use life tickets to revive yourself in battle but there is a limit of 5 per day.
Free Quests are events that allow you to play as guest characters for a limited time. Guest characters usually have different skills compared to their original counterparts without any influence from the player's main party members or hero's ability so free quests are usually easier than normal quests.