I will try to answer directly original questions.
Basic difference is in their purpose and connection to power system.
VT will be connected in parallel, and thus always see voltage of the system (rated and maximum expected, defined by voltage factor – 1,5 or 1,9). Current through VT primary winding will be small, caused by its magnetizing current and load, which is relatively small (up to 200 VA).
CT is connected in series in power system, carrying rated and fault currents and transmitting them to secondary winding in expected ratio.
Purpose of instrument transformer is to give accurate image of primary voltage or current to secondary winding and circuit. Therefore, their error must be minimum.
CT and VT have same equivalent diagram as power transformer. Difference is in value and significance of its elements.
Error of VT is caused by horizontal branch of equivalent diagram, because VT load created current which makes voltage drop on those elements. Magnetizing branch (vertical branch) is fed by power system and does not influence secondary voltage. At least not directly. Magnetizing current creates additional voltage drop on primary winding resistance and leakage inductance. Mentioned voltage drops create voltage error and must be minimized in VT.
So, answer No.1: error in VT is generated by equivalent diagram horizontal branch. Saturation of VT is not an issue because it is designed not to be saturated at all voltages expected and defined by power system (defined b voltage factor).
In CT, situation is opposite. Current through primary winding is defined by power system. Current error is caused by magnetizing branch (part of primary current goes to magnetizing branch instead to secondary winding). Burden connected to secondary winding defines voltage which needs to be induced in magnetizing branch and thus current required for magnetizing. When voltage or current of magnetizing branch reaches saturation (excessive burden or excessive primary current), transformation ratio of CT changes - its error increases significally, because magnetizing current increases significally.
Answer No. 2: Magnetizing characteristic of CT defines its accuracy – current error, as well as important CT values – Accuracy limiting factor for protection CT, and instrument security factor for measuring CT. Both are caused and controlled by CT saturation. Explanation and details you can find in any power system engineering handbook or relevant CT standard - IEC 60044-1.
Note: I am from IEC world and definitions I mentioned refer to IEC practice. US/IEEE world has its own definitions of same items.
Sorry for long answer, hope you find it helpful helpful. Check literature for details and drawings, much easier to understand.
