+1 on the Apogee Peak of Flight Newsletter linked above, that explains it well.
To summarize very briefly here: the two methods are based on different assumptions. Barrowman equations are based on the rocket flying nose first into the wind, at zero-deg angle of attack. The cardboard method is based on the rocket flying sideways to the wind, at 90-deg angle of attack. The cardboard method is more conservative, if you use that to determine the balance point of your model, yes it will be a stable design, but more stable than is needed, i.e. it may require you to add weight that you don't really need to the nose, which might not be so important just for fun recreational non-contest flying, but as a consequence it also may tend to weathervane into the wind much more than desired.
This is a good explanation.
These are two different methods, so they're going to indicate two different points. The cardboard cutout method is actually the center of lateral area, which coincides with the CP in the worst-case scenario, that being, at a 90 degree angle of attack to the direction of flight (IOW, rocket nose pointing 90 degrees to the direction of flight, or "flying sideways"). In reality, you should never see such an enormous angle of attack, and since CP is a FLUID point, NOT a FIXED point (it moves around based on the angle of attack, the airspeed, air density, and a lot of other factors in flight) technically speaking this is the furthest forward the CP should move in flight, because a 90 degree angle of attack is the "worst possible scenario" (and will move the CP the furthest forward). IOW, the *real* CP should always be BEHIND where the cardboard cutout method indicates... SO, if the rocket has at least a 1 caliber stability (one body diameter between the actual measured CG and the cardboard cutout CP, it should (theoretically at least) always be stable.
Now, the Barrowman equations are just ONE way to calculate the CP... they rely on certain simplifying 'assumptions' that might not apply to the rocket design, especially when it gets more complex and deviates from the "3FNC" designs... The Barrowman equations are the starting point for all the other methods, which then use their own specific assumptions and "refinements" to "more accurately" figure out the CP (more closely estimate it). The RockSim method (in the RockSim simulation program) makes certain refinements and different assumptions and methods to determine the CP; one can see this by switching between the three methods (cardboard cutout/center of lateral area, Barrowman method, RockSim method) and comparing the calculated CP points from each. The Barrowman and Rocksim methods are usually pretty darn close to each other, depending on how 'different' the rocket design is from the "3FNC" type rocket shapes assumed in the Barrowman equations.
It's good to compare methods.. if you have the tools available to you, avail yourself of them. CP is something that can never *quite* be nailed down to 100% accuracy anyway, since it moves around depending on the dynamics of the rocket in flight. That's why regardless of what method you use, they are all still basically, at heart, approximations.
Later! OL JR