The first advantage is the elimination of war (isn't that enough?) It is clear and straightforward that this would be the case; one army cannot have a war by itself.

The next advantage is the elimination of nuclear bombs and all other weapons of mass destruction. It just would not be appropriate to keep them anymore. It could be the first job of the army to see that every one of them was disposed of, and that nobody developed and manufactured them any more.

The army could be greatly reduced in size; this means that the cost in maintaining it would be far less than the total cost of maintaining the numerous individual armies that exist today; which in turn means more money for all the world's other problems.

The army would be in a very good position to deal with international terrorism; it would already have a ubiquitous presence in the world and could inherit a multitude of intelligence networks.

The army could police and maintain international borders, thereby appeasing man's inherent territorial nature.

The army could undertake the job of removing all the thousands of land mines and other live munitions that are laying about in some countries waiting for some child or other innocent person to step on them.

The army could enforce international law and other dictates made by the United Nations. At the moment the one shortfall of the U.N. is that it has no impartial agents to enforce its rulings.

The army could exist as something to be relied upon in times of natural disaster and crisis, and as a resource supporting any humanitarian causes needing assistance.

The army could remain as the foremost institution for the development of human character, fitness and abilities, ( one possible good thing to come out of a history of warfare).

Under such an arrangement, countries would remain free to be able to pursue and develop a diverse range of systems of government and policies. 

If it ever turns out to be true that we are under threat from intelligences from other worlds, (we have been watching cable TV)  then as Ronald Reagan pointed out some time ago:  one united world army would be far better disposed to dealing with that threat than a lot of disparate armies under different commands.