The major difference between the two parties at first was the fact that the Democrats supported Pres. Andrew Jackson while the Whigs formed specifically to oppose him. This was really all that bound them together at first.
In general, the Whigs tended to support a stronger Congress and a weaker presidency. They were also in favor of the sort of internal improvements that Hamilton had wanted. In this sense they were like the Federalists had been -- they wanted the government to actively work to help improve the economy. This is in contrast to the Democrats who were suspicious of economic elites (as with Jackson's war against the bank).
As this implies, the Whigs (like the Federalists) were more of an elite group while the Democrats were more of the common people.
The Whigs did not last long and broke up on North-South lines over the issue of slavery.
several events and issues helped destroy this newfound spirit of political unity and fostered the return of the two-party system. First came the financial panic of 1819, which cultivated newfound suspicion of neo-Federalist economic policies. Next came the contentious election of 1824, during which John Quincy Adams gained the presidency despite General Andrew Jackson's winning the popular vote; this event caused many ordinary Americans to feel a new sense of distrust toward the political system. However, the true death-blow to the Era of Good Feelings came with the election of Andrew Jackson in 1828. A staunch "man of the people," Jackson detested anything that even remotely smacked of corrupt bureaucracy or elitist aristocracy; he espoused an extreme form of laissez-faire economics and advocated policies that would democratize the political system and give power to ordinary people. (Jackson, incidentally, dropped the "Republican" from "Democratic-Republican"; his party simply became known as the Democratic Party.)
Somewhat paradoxically, Jackson implemented his democratic philosophy by seriously stretching the bounds of presidential power--a very undemocratic way of acting. From his war against the National Bank to his authoritarian handling of the nullification crisis in South Carolina, Jackson's actions quickly led to his opponents' dubbing him "King Andrew." They also led to the rise of a strong political opposition, eventually named the Whig Party. A diverse coalition of anti-Jacksonians, the Whigs tended to advocate greater federal authority and more governmental control of the economy. They supported the National Bank and a policy of protectionism, and they contemptuously considered the Jacksonians' "democratic" philosophy as mere class warfare rhetoric. An 1845 article in the American Whig Review said of the Democratic Party, "By its stupid cry of aristocracy, it has sought to engender the most unnatural war between those natural allies, the poor and the rich..