There are non-linear equations that can be used to model most things more accurately.
When you start with the fundamental equations describing the ideal non-linear capacitor, such as the ones listed above, you can then add various things like noise, hysteresis, non-linear resistance, non-linear inductance, RF characteristics, etc. Each addition gives you more accuracy but also requires more input parameters to describe those characteristics.
My guess as to why the curves are different is the non-linear capacitor includes some of these effects. Most likely the noise and inductive effects as these are more commonly modeled.
Also, the measurement of these introduces errors. There are quantization errors in the ADC, loading effects due to the probes, etc. RF effects being picked up, etc. The list goes on and on. The main thing to get out of is actually that the graphs are pretty close. The main differences seem to be "noise". In 99.9% of all electronics, you'll be ok making the assumption that capacitors are ideal/linear(within the SOR).
I would point out that these non-linear effects are generally not required and not desirable to model. 1. They slow down analysis significantly and are not always stable. 2. They don't add much to the precision of the analysis. 3. Actual components are more variable so one never will be completely accurate. 4. Most components function quite close to the linear model but vastly different to the non-linear model. (e.g., because of noise, no two components ever function exactly the same)
5. Measurement of actual components require using real measurement devices. These devices are also inaccurate so they add to the inaccuracies. The linear model and some of the non-linear models are derived from first principles(math + physics) and hence are "true" and can be relied on.