Unfortunately, there might be a chance for a bit of inaccuracy here from the Miami Herald news article.
The periodic table lists the elements in order of their atomic weights, but when Mendeleev was classifying them, no one even knew what was inside these tiny things called atoms.
While it is true that, historically, Mendeleev originally arranged the elements with respect to each atom's atomic weight (since no one knew that was inside these atoms at that time), the periodic table that we have now lists the elements in order of their atomic number, i.e. the number of protons in the element. This is because we now know that an element of a particular atomic number may have several different isotopes (atomic weights). So the atomic weight is not a unique number for an element, but atomic number is. That is why the period table is arrange in order of the element's atomic number.
In any case, Happy 150th Year, Periodic Table!