>To clarify, I guess what I meant by “distinguish” is whether or not different isotopes behave fundamentally differently as far as biochemistry is concerned

I would maintain that if the isotopes are incorporated at different rates (as they indeed are), then they behave differently by definition. I'm not sure what "fundamentally" means here - if you mean "substantially" in the sense of having biological relevance, then I would say no, they do not. But "biological relevance" itself has no objective definition. I could say that they form the same sort of chemical bonds, but that's not actually entirely true, just mostly true, of isotopes.

>or is it just a side effect of getting carbon from CO2 in the air versus carbonic acid/carbonate in the water?

I should specify that CO2 ultimately comes from the atmosphere in both cases. The difference is the carboxylation reaction in C3 plants uses carbon dioxide directly, whereas in C4 plants, CO2 first reacts with water to form bicarbonate before being conjugated to an organic molecule. The relevant factors for fractionation are therefore the diffusion rate of 13CO2 and 12CO2 in the gaseous state, and the preferences for the relevant enzymes for each carbon isotope.

The underlying physical principles are the same here as in the case of neutron-free hydrogen vs. deuterium. The difference is just one of degree. Carbon-13 is 8.3% heavier than Carbon-12, while deuterium is twice as heavy as hydrogen. Moreover, hydrogen atoms (of all isotopes) are commonly transferred between compounds individually, whereas single carbon atoms do not appear in biological reactions. (In the specific case of carbon fixation, the carbon makes up a minority of the mass of the molecule that actually participates in the reaction.) The discrimination between hydrogen and deuterium in chemical and physical processes is therefore as high as it could possibly be for stable isotopes, and the differences in rates between them is therefore maximal compared to other elements. These discrepancies in rates, which differ in relative magnitude direction for different processes, are enough to upset the balance of biological systems if they are supplied with too much deuterated water. However, I also can't say for certain that a biological system supplied with only 13C wouldn't suffer a similar fate. After all, we're comparing the partitioning of naturally-occurring isotopic ratios of stable carbon isotopes to the extreme hypothetical of exposing an organism to pure heavy water.