Some poor arguements I can think of are that the table itself is huge making it a pain if you're only interested in a certain group of elements (though most related elements are listed very well by the table), and doesn't really show any relation between certain properties of transition metals (though it shows some others, and it could probably be adapted to show more, like I said, poor arguement). An arguement against chemical symbols for equations is that some require two letters which can be confusing in equations (most equations use one letter and sub/superscripts), and some use the same letter that you may use for a constant/variable in chemistry/physics so it can get messy in essays. It also removes an understanding from what happens in a chemical equation, conserving elements is easy and arranging them in the logical way isn't much further, and this takes away the need to understand the process behind what is happening (i.e. the stuff with electrons flying between things and bonds forming, any intermediate reactions etc. I don't really know beyond my memory of A-level chemistry). Some more examples of the above disadvantage would be Fe - Iron, Na - Sodium, K- Potassium, Au - gold, Ag-silver, these abbreviations don't seem to show any relation to the English word.
In general though they're useful things, I'll never write out full names for elements/compounds again.
Also, sulfur? Screw that, it will always be sulphur.