How is the modern periodic table different from the table Mendeleev proposed?
In Mendeleev's day (c. 1870), atomic number was not known. (It would be discovered by Henry Moseley in 1913.) In order to put the known elements into some sort of order, Mendeleev used the atomic weight, which was known for many elements in his day, but not with the precision we know these values today!
When Moseley showed that atomic number (the number of protons in the nucleus) gave a better means of listing the elements in order, it forced a few instances where elements "traded places" in the table. Iodine and tellurium were switched, as were nickel and cobalt.
A second difference is that none of the noble gases were known in Mendeleev's time, (they were discovered in the mid-1890s) and the periodic table he proposed had chlorine followed by potassium for example.
Also, of course, Mendeleev's great triumph was to leave spaces in the table for a number of elements that had not been discovered by 1870, including germanium and gallium. Today, all naturally-occurring elements are known, and the spaces have been filled in.
The table has been proposed in a wide variety of forms, besides the familiar "long form" that graces the walls of most high school Chemistry classrooms, and Mendeleev's original table would have looked strange to a modern student of Chemistry, but the fundamental form we now today was largely in place by 1871.