Using mmol/L helps with revealing some useful relationships between groups of analyzed substances while using mg/dL might obscure those relationships.
I wasn't able to find original proposition to switch from mg/dL to mmol/L which should be in "Quantities and Units in Clinical Chemistry. Including Recommendation 1966 of the Commission on Clinical Chemistry of the International Union of Pure and Applied Chemistry and of the International Federation for Clinical Chemistry. Munksgaard,
Copenhagen" by Dybkær and Jørgensen (1967), but by knowing where it should be helps tracking later publications discussing the change. There are also some other publications from Dybkær like for example "Quantities and units in clinical chemistry" (1969), where he shortly states:
(...) it is recommended, as far as possible, to use the 'chemical' basic kind of quantity 'amount of substance' with its basic unit 'mole' (symbolized mol) instead of 'mass' and mass units.
This seems to suggest, that mmol/L is better because it's more suitable to proper chemical, laboratory work. This is too vague. Some really interesting comment is made in "SI Units: One Step Closer" (1985) by Lehmann and Beeler:
The rationale for the introduction of concetration units based on amount of substance, first proposed internationally in 1967, was that use of those units might reveal functional relationships between analytes, potentially useful in in the interpretations of results, but obscured when concentrations are expressed in mass units. (...) With further advances in understanding molecular bases of disease, and with the introduction of new technologies into the clinical laboratory (...), we may expect increasing use to be made of such biochemically nad physiologically related groups of analytes in order to derive new diagnostic and prognostic information from laboratory tests, and thereby to increase the usefulnesss of laboratory data. Reporting results in amount of substance units will most likely contribute to this process.