Here I do show the differences between the two datasets (formally the same dataset). The corrections are within the accuracy of the data (+/- 0.01 °C), so no problem for that; but they are (almost) sistematic over different periods (down from 1915 to 1960; up from 1960 to 2015; many up-some down from 1880 to 1910). That produces e.g. an additive 0.02 °C in the slope between 1915-1960 and 1960 2015.
The need to change data within the same month seems strange, because the users cannot say for sure which dataset they make reference to.
The same, but with a different distribution of sistematic corrections, happens for the Global Ocean Data.