The biggest reason to user royal arms are the stat boosts. A very viable and highly suggested strategy is to equip 3 royal arms and one actually "used" weapon or spell, to maximize the stat bonuses. Personally, I use the sword of the father and Axe of the conqueror in two slots as they increase str, which increases Noctis' dmg per hit. I then use the star of the roue just to have a ranged option, but Sword of the Wise is a good option for well rounded stats, and shield of the just is great for HP and defense, but it takes away stength. If your have built as a mage, there are arms such as the Scepter of the Pious that give a bonus to magic, and even uses magic for its normal attack damage.
They are also useful against enemies who are resistant to elements and/or weapon types. Royal arms all have no elemental affinity and also do not "count" as any weapon type for damage resistance. So if an enemy has two handed sword resistance, you can use the Sword of the Tall, or Axe of the Conqueror royal arm, you will bypass that resistance.
As a side not, if the enemy has an elemental resistance you want to exploit, but has weapon resistances that dont make you happy, you can use Ignus' Enhancement ability on your royal arm to give it the elemental damage bonus while still avoiding the weapon resistance of the enemy.