Samuel, as a priest, had the right according to God’s established order, to offer sacrifices on behalf of the people. But Saul, as king, did not have that right. And when Saul didn’t wait for Samuel to perform the sacrifice, but did it himself, God rejected him as King.
This mistake may seem trivial and pardonable on the surface, but Saul’s actions, as the first king of Israel, put his own thinking ahead of God’s established order. And that would be a bad precedent to set at the beginning of Israel’s monarchy.