Back in the days, direct democracy is where the will of the people is translated into public policy directly by the people themselves. And republic democracy was held by those eligible to vote, while the political power is exercised. How voting is different in a direct democracy compared to a representative democracy?
The difference is what is on the ballot.
In a Representative Democracy a ballot is a list of names of people who might make good leaders, who will then decide what society ought to do. Generally those people are then given the authority to enforce any decisions they make.
In a Direct Democracy the ballot is a list of things it might be desirable for society to do. This group decision is then binding on whatever leaders exists, and to go against the will of the group probably has a proscribed punishment like usurpation or treason.
In practice the line between person and project blurs a little since issues important enough to bother with and having few enough plans to make a reasonable vote normally have individual champions or leading thinkers who make the explanation to the voters what voting for that plan means, and nobody wins votes without some hints about what direction they are planning to go in.
With parties, especially ones with detailed manifestos, it gets downright unclear. You might vote for some not quite specific people to have broad discretion but who have been selected based on their agreeing on some specific issues.