Is the HP206C really so precise?
Accuracy is not resolution !

Accuracy means just that, how accurate is the reading. For this sensor:

Pressure Absolute Accuracy (0 C to 50 C): -1.5 to +1.5 mbar 

That means if the actual pressure is say 1000 mbar, then the sensor is guaranteed to report a value between 998.5 mbar and 1001.5 mbar.

Resolution is different, it is about the number of digits you see so

1000 mbar vs 1000.01 mbar

On the 1000 mbar reading the resolution is 1 mbar, a difference of 1 mbar is needed to get to the next lower or higher reading.

On the 1000.01 mbar reading that step is only 0.01 mbar so it has a 100x higher resolution.

That 1000.01 mbar doesn't have to be more accurate.

For example,
Sensor A showing a reading of 1000 mbar, it has a resolution of 1 mbar and an accuracy of +/-0.5 mbar

Sensor B showing a reading of 1000.01 mbar, it has a resolution of 0.01 mbar and an accuracy of +/-1 mbar

Obviously sensor A is more accurate, sensor B just suggests that it is more accurate (by showing more numbers) but it isn't.

You will however be able to see smaller changes with sensor B.