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 the 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 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 a 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.
I wrote this long answer to a question which was deemed "duplicate" but really asked a more specific question, why there are the gaps in the periodic table etc.
The elements are organized in groups (the columns) because the elements in each group have very similar chemical properties.
For example, the noble gases are inert: today, we know that it is because all the electron shells up to a rather big gap are occupied, so there is no free electron that can create bonds with other atoms; and it is not easy for the atom of the noble gas to adopt an additional electron, either.
On the opposite side, the group starting from hydrogen and lithium contains highly reactive elements because they have exactly one electron on top of the full shells above.
If one considers elements with electrons in the shell $1s$ only, there are only two elements, hydrogen and helium (the maximally reactive; or the maximally inert), and nothing in between. There is nothing in between because the shell $1s$ only contains two "slots" for electrons, so the number of electrons may only be $0,1,2$ (zero is not an atom).
However, as we go down the table, towards elements with a higher number of electrons (and protons in the nucleus), there are more possibilities. For example, in the second row, we have 8 elements. Why? It's because the shells $2s$ and $2p$ contain two and six states, respectively – eight in total.
So if we want to have at least one electron in the $n=2$ shell, there are eight options, from $1$ to $8$ electrons. Again, the element with $1$ electron in the $n=2$ shell is analogous to the hydrogen, and the element with the maximum number $8$ electrons, i.e. neon, is analogous to helium because the shells are fully occupied.
But unless the first group which only consisted of hydrogen and helium, there are 6 new elements in between, beryllium, boron, carbon, nitrogen, and fluorine. Their occupation is something in between the minimal and maximal. Those have no counterparts among the simplest elements that only work with the $1s$, $n=1$ shell because there are simply not six extra integers in between one (hydrogen) and two (helium). That's why the gap appears.
When we work with the more complicated shells, we always reveal some new "intermediate" elements that had no counterparts in the simpler periods. That's why one needs to create increasingly wide "gaps" in the upper periods – because the lower periods have some additional "intermediate" elements because there is a greater number of ways to fill more complicated, larger shells.