What does it mean to call something Leviathan in comparison to others?
It is a reference to the philosopher Thomas Hobbes' magnum opus, Leviathan. In that work, Hobbes argues for a powerful, far-reaching state -- a leviathan, so to speak -- to curtail what he sees as man's base and destructive instincts without the hand of a strong sovereign to guide him; it was Hobbes who famously called the natural state of Man "solitary, poor, nasty, brutish, and short." Hobbes did name his book after the massive beast in the Bible, but he used the reference to the Beast's massiveness in a metaphorical and positive sense to emphasize the key idea in his theory of the social contract.
In other words, what The Economist is saying is that Mexico is better off than Guatemala and Honduras because their state has capability to protect against the base state of Man. E.g., against the petty thievery, illegitimate violence, and capricious laws that plague poor, underdeveloped countries, the very same things Hobbes abhorred in his book. As you can you see, the excerpt highlights Guatemala's lack of state control over heavy arms and prisons, traditionally one of the fundamental prerogatives of government -- it clearly does not live up to the ideal Hobbes espoused in Leviathan. Mexico, as weak as its government is, is closer.
As I have always understood it, the clue is that a strong state is a necessary evil. The Biblical monster Leviathan is a metaphor for this strong state. Yes, a strong state will cause corruption, abuse of power, and a general waste of resources, as well as injustice; but a weak state amounts to or leads to anarchy, which will cause much, much more waste and injustice. Let one King have a huge, wasteful palace, requiring burdening taxes, and kill whomever he likes; this is still to be preferred over dozens of warlords with similar courts and habits.
Moreover, with one King, at least the nation will be spared civil war, which is the greatest of all evils, with its tremendous loss of lives, trade, and resources. Hobbes had experienced the English Civil War (1642–1651), which probably influenced his conclusion. Civil war is a form of anarchy, the worst of the worst to Hobbes.
In the case of Mexico, the author means to say that, yes, the Mexican government is to a large extent corrupt, unjust, and wasteful; but at least it is strong enough to prevent anarchy, which the equally corrupt governments of Guatemala and Honduras are not. There is nothing between Leviathan and anarchy, and at least Leviathan provides some degree of stability. Of course much could be said against this position.