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I am following the diagram that is attached below:
Now, say I have the following circuit:
And I apply this theorem and end up with 
Then, even though the current was the same in Circuit B for R, R1, R2; it is generally different for R than for R1 and R2 in circuit A. Why is this?
It's because the theorem about equivalence of Thevenin and Norton sources states only that the currents and voltages in the external circuit (i.e. anything to the right of points (A) and (B) in your circuit) are the same.
Not the voltage across or currents through the series/parallel resistors of the equivalent Thevenin/Norton sources. Those resistors belong to the sources.
I.e. in this case only currents/voltages at R1 and R2 are the same; not currents/voltages at R.
This fact that current/voltage at R can't be the same is obvious if you assume your external circuit to be e.g. an open circuit (or a short circuit):
if the external circuit is an open circuit the current through R of a Thevenin
source is 0; the current through R of an equivalent Norton source not.
