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.