Wands, Staffs, Focuses, and General Spell-casting

Chaos is a dangerous, unpredictable element that takes years to acclimate to and more to even use. Tools or special knowledge is needed to ever control what is known as magic.

Wands
Wands are custom crafted, specially made focuses with chaos-imbued wood and a blue gem that manages the chaos in a mage's body. A mage can hold a wand to cast spells, but it only needs to be on their person. Moreover, if a wand breaks, the caster will lose more or less half their spell slots and would be rendered unusable until their wand is repaired.

Staffs
Staffs are more for convenience for witches; they absolutely do not need a long wooden stick, but it certainly is helpful for them to use one to focus their thoughts. They can use it to lean upon, to drive away wild animals from a distance, and to, in general, use as a long stick.

General Spell-casting
Witches manipulating chaos comes more naturally for them, as if it was "2+2" while mages have to take Calculus III to figure out the Y variable of a spell in which X is the terrain, and Z is the distance of a target, and you have only your inefficient calculations from Eramu to work upon. While Witches have a wide array of spells they may learn, mages only have their specific school to work and graduate on. Similar to learning a skill, a mage can take 4 hours to learn a spell eventually, granted they work on it for at least a week.