Independent generic features (Insert | Feature | Generic) have values that remain the same, regardless of which alignment they are referenced from (therefore considered "independent"). Their location in 3D space changes as the alignment changes.
For an example in 2D, consider the same example of 2,2 (as shown in the Alignment Dependent example) but this time with an alignment independent generic feature. If you ask for the x and y values relative to the alignment, the object will report 2,2. If you ask for its values relative to the origin, the object will still report 2,2. The set 2,2 relative to the alignment is located at 7,7 relative to the true origin. The set 2,2 relative to the origin is exactly that: 2,2. Thus, the point seems to move in 2D space.
The independent option was added so that a generic feature named XAXIS, for example, could always be used as an input feature to a construction or a datum to a dimension, and it would always have the values of 1,0,0 independent of the current alignment.