What are the different diagrams/tables used to organize the elements other than the Periodic Table? What are the advantages/disadvantages of each?
One periodic table that apparently physicists find useful is ADOMAH periodic table, for its usefulness is finding electron configuration of a particular element. More information at this guide. A disadvantage to this table, along with any new table configuration you have not experienced, will be the lack of knowledge in the trends on your new table. There are many spiral looking tables, but the trends such as change in atomic radii, or electronegative characteristics, might not be as obvious, and you might have to find a guide or relearn old tricks to methods you would learn in your first year of chemistry. At least the ADOMAH Table is built in such a way that that these trends should still be recognizable, although clearly not the same as the normal table you find in any high-school science classroom.
Of particular interest to me are dynamic periodic tables. These are tables that are basically computer applications which will change according to what you are looking for. One outstanding example is the widely popular ptable.com which will maintain all the common trends you probably know already, and dynamically show you various states of the elements when changing the temperature. I learned from this table that rhenium has the highest known boiling temperature, while I thought it was tungsten (although tungsten has the highest melting temperature of the transition metals). And better yet, carbon has the highest melting temperature of all elements, very cool table! The disadvantage is obviously that you cannot carry it around in your notebook without some flavor of mobile computer and/or internet connection.
I am sure there are many other great tables still to be found, but I hope you have learned something new from my post.