The main argument between these two factions was over how much power the federal government was given in the new constitution. The Antifederalists thought that the federal government was too strong and therefore they opposed the ratification of the Constitution.
The Antifederalists were generally happy with the Articles of Confederation. They did not want a strong central government. They felt that such a government would be too much like the government of Britain that they had just cast off in the Revolutionary War. They felt that such a government would be too likely to take away the rights of the people.
By contrast, the Federalists wanted a stronger national government. They felt that state governments were too democratic. They felt a strong national government would do a better job of doing what was good for the country as a whole.