The late 1800s, defined, perhaps, as the time from the end of Reconstruction until 1900, were not a time of very many major political differences between the two parties. The two parties each had roughly half of the support among the people and elections tended to be very close. Much of politics involved local issues rather than national issues. That said, there were a few major issues at play in those days.
The best-known of these issues was the issue of gold and silver money. The Republicans were strongly in favor of the gold standard. The Democrats were more concerned with helping the common people. They tended to support the farmers, who wanted the coinage of silver money was a way to make it easier for farmers to pay their debts.
There were also debates over the proper level of tariffs. As the party that was generally representing the “haves” in the country, the Republicans were in favor of tariffs. They felt tariffs would help American manufacturing companies. Democrats were more likely to be in favor of lower tariffs.
Another issue that was relevant to some degree was government regulation of big business. Again, the Republicans were generally in favor of big business and did not want to regulate it. The Democrats wanted to do things like preventing railroads from charging “excessive” prices to farmers who had to use them as ways of getting crops to market. By the end of the century, the Democrats were also pushing for anti-monopoly legislation.
These were some of the most important issues in politics in the late 1800s.