How can cane sugar sucrose be different from beet sugar in regard to carbon-13 content?
The differential uptake of different isotopes of carbon is covered well at:
https://en.m.wikipedia.org/wiki/Fractionation_of_carbon_isotopes_in_oxygenic_photosynthesis
Plants have a lower proportion of C-13 than atmospheric CO2.
The two mechanisms that cause this are the reduced diffusion speed of the heavier C-13, and an energetic difference in the chemical reactions that favours C-12.
In sugar cane (C4 photosynthesis) the carbon dioxide is captured in an additional step compared with sugar beet (C3 plant).
This capture step in C4 plants (using phosphoenolpyruvate carboxylase) is less exclusive toward C-13, so the carbon in these plants has detectably more C-13 than C3 plants.
This subsequently affects the isotopic proportion of all the carbon in the plant