"The American Scholar" (Emerson) and Arnold's thoughts in "Democracy" are similar in their that both thinkers believe that individuals and society should aspire to high moral and intellectual standards.
However, they have different ideas about how individuals and the society as a whole can achieve those high standards. Arnold looked at the rise of democracy with skeptical optimism, emphasis on the skeptical. Arnold was not opposed to the liberation of the lower classes which democracy promised to bring. But he did think that with the increase of democracy, the aristocracy (nobility, upper class, and high intellectual culture) would begin to lose its influence on the masses. Therefore, in Arnold's thinking, society would become more liberal via democracy, but it could devolve without the aristocracy to provide examples of the highest moral and intellectual standards. In other words, Arnold was in favor of people of all classes being free but he thought there must be a central authority (something like the aristocracy) to guide all classes by example. This would be somewhat of a combination of democratic ideals but with a central authority/State to guide and instruct society. This is more of a top to bottom structure of social instruction.
Emerson's model was more of a bottom to top. He believed that individuals did not need a central authority of morality and intellectualism. He believed that each individual had the creativity and genius in him/her. It will take effort and will to tap into that genius, but Emerson thought it could be achieved (through learning, activity, individual work, and work with others). This concept of Emerson's is most cogently described in his essay "Self-Reliance." If each individual did his/her best to seek the wisdom of high morals and intellectualism, the society as a whole would improve. The collective of individuals working independently and then together would form a united society that would improve from the ground up.
Another sign of our times, also marked by an analogous political movement, is, the new importance given to the single person. Every thing that tends to insulate the individual, — to surround him with barriers of natural respect, so that each man shall feel the world is his, and man shall treat with man as a sovereign state with a sovereign state; — tends to true union as well as greatness. (Emerson)
Arnold even called this focus on the lower class and self-empowerment as "Americanized." He didn't think this would lead to a better society (from the ground up). Rather, he believed a society would still need an upper class or central authority of some kind to "culture" the lower classes. Emerson wanted people to think for themselves and he believed that individually, people could attain such high standards of morality and intellectualism without a central authority telling how to do so.
This was, and still is, not just a question of how much a government should instruct its people. It has to do with all of culture: the arts, morals, academics, etc. Emerson was a champion of the individual. Arnold was a champion of the individual but also of a upper crust/class to lead by example.
Thank you! This really helps. Reading Emerson and seeing his ideas was much easier than the ideas of Arnold.