Lawful characters are more inclined to use logic and reason to solve problems, as opposed to "gut feelings" and emotions. Lawful characters see a society with rules and order as a positive thing, but these rules can always be improved upon. Lawful characters value teamwork and truly believe "the good of the many outweighs the needs of the few". Concepts like justice, honorable and moral conduct, and "not asking what society can do for you, but what you can do for society" are central to Lawful characters.
Of course, while Lawful Good characters try to live up to these ideas, with their belief in a better world, Lawful Neutral characters are more concerned with equity and fairness. The law must benefit the rich and the poor equally, and no deviation is allowed- you can fix the system, but to step outside of it will cause collapse.
Lawful Evil characters are the ultimate rules lawyers- while they too see social order as important, they have no problems exploiting loopholes and making sure the law benefits their interests above all others.
I want to point out that non-Lawful, even Chaotic characters can espouse some of these traits, but overall, they tend to be independent, self-willed, and free thinkers, who think that if the law is not a benefit, then perhaps it's time for a different kind of law.
Lawful characters want change to be slow and methodical, with the ultimate goal of perfection, an order so perfect that change will no longer be necessary.
Chaotic characters want change now, and the devil take the hindmost.