In theory, both live IK and offline IK use the same quaternions.
In the case of live IK, the quaternions are at double precision (or some other high-precision format) and therefore the calculations are slightly more accurate.
In the case of offline IK, those quaternions have been saved to a file, which has reduced their precision. When these quaternions are read from file and used for IK, they are rounded to some extent.
This leads to usually very small differences in IK output values. These differences are negligibly small and are usually of no consequence.