The TL;DR of the answer: sorcerers are contemplative, wizards are willpower driven.
Sorcerers use the crystal to feel the magic surrounding them, see which entity has to offer what kind of magic and ask a number of those entities for certain combinations and sequences of magic ether releases to achieve a certain effect in a certain place.
To do this, they use the crystal ball:
- to reflect the magic landscape around them though the ball
- to convey his asking to the magic entities around
- to focus the offered magic in producing the effect they want
For all of the above to work, the crystal has to be large, be it only to accurately reflect the magic reality around.
The process is surely going to take more time (even if it will be pretty fast for highly skilled sorcerers, who a "speaking" to the magic world fluently and perceive it in an instant), but are unlikely to produce imbalances in the magic around them and induce decays to to magic depletion.
It is said that high sorcerers can even read the past and intuitively feel the future of a place based on the flow of magic they perceive it there. This is how the legends of divination was created around them, but the temporal dimension of the magic is handled better by other branch of magic... still using crystal ball but even listing the fundamental differences is beyond the scope of this lecture)
By contrast, wizards don't waste time to ask for magic flows to be offered, they demand it and, when they feel it's needed, they forcefully appropriate it by the use of spells. They are living here and now, reacting swiftly and taking fast (and not rarely hasty) decisions.
At high level of proficiency, wizards won't affect the balance of magic in the surroundings by requisitioning what they need - at least not when they can avoid it. They don't do it from an empathic or feeling based ethics, their approach is strictly pragmatic - those that did that in the past didn't last long in confrontations, no matter how strongly willed one is, one can't take anything from a place they depleted earlier.
Now, sensu stricto, it is improper to say the wizards do not use crystals to control the flow of magic. After all, not every material is appropriate to make a magical wand from. At microscopic level, no matter the wand overall material, there are strands of crystal that channel the flow of magic. The already known requirement of a certain degree of fitness between a wizard and his wand has explanation in the way the configuration of these strands (and the interplay of the strands with the embedding matrix) feel for the prospective owner. Where for most of the other wizards feel the constraints of the matrix and bending or torsion of the crystal strands as impeding the magic flow, the prospective owner feels opportunities of flow control and modulation of the wave of this flow.
the ability of a wizard to take control on the wand of a wizard foe they defeated in a fight stems from the fact they felt and internalized how the opponent used it - it is a necessary condition to withstand and repel the spells and the burst of magic shaped and directed as attack vectors.
S/he won't be able to have the same proficiency using the captured wand as when using their own, but still the level of control over the captured wand is 1 or 2 levels of magnitude better then any other still-alive wizards