There are two ways in which to interpret this question.
First, the question could mean something like “what did these two groups disagree on?” The two groups disagreed on two main issues. One of these issues had to do with what the US economy and society would look like. The Federalists wanted an economy that was diversified and they wanted to begin to have manufacturing. They accepted that this would lead to some stratification in society. The Democratic-Republicans wanted an economy in which practically everyone was a small farmer. They did not want to have any stratification. Instead, they wanted a society where everyone was equal because everyone was self-sufficient.
The second issue had to do with foreign policy. During this time, England and France were fighting in the Napoleonic Wars. The Federalists sided with Britain. They liked Britain’s more conservative system of government better. The Democratic-Republicans sided with France. They liked the system that the French Revolution had created.
Second, this question could mean “why did the two sides disagree on these issues?” That question is harder to answer. It could have been because they had different economic interests. The Federalists were more likely to be townspeople who relied on things like trading. The Democratic-Republicans tended to be farmers. This could be why they felt as they did.