The weapons you'll get for completing scenarios will contain vary in base damage and skills. To tackle harder enemies and Hard Mode, you need to find and use better weapons. Some Characters in the game can wield two types of weapons, each their own set of combos and SP attacks. You can find new weapons from chests or dropped by "leader" enemies. (Larger enemies that have life bars and you can lock on as well)
More weapon types for each character can be unlocked via in Adventure Mode.
If you have a lot of unneeded Materials to spare, you may want to use them to create Mixtures before you start a scenario. This is available in the Bazaar's Apothecary.
Each type of weapon has an upgraded version. For example, Link starts with a Hylian Sword. All Hylian Swords you'll find will have the same base damage but will have different number of slots, skills, and "stars" (rank). The next tier weapon is called a Knight Sword, and so on. There are also weapons with Elemental Attributes. Equipping the recommended element for the right scenario can make a difference.
Stars give bonus damage to the weapon's base damage. Each star adds 10% bonus damage so if ever you have identical weapons with the same power and slot/skills, check which of them has more stars.
If ever you find a Sealed Weapon, it will be a new type of weapon or an upgraded version of what you currently have.
Each character can only keep up to 10 weapons. If ever the weapons they collected from the last battle exceeded the limit, you'll be forced to sell the unwanted weapons.