The primary advantage of electing a candidate based on the popular vote rather than an indirect (electoral) vote is the equality of each individual's voice during the election process. In the United States, the current presidential election relies on awarding electoral votes based on state voting results rather than accepting the popular vote as the final result.
In recent memory, the most controversial election result was the presidential election of 2000, in which Al Gore would have been awarded the direct election by over 500,000 votes. Some experts point to the electoral college process as the reason for voter apathy in the United States due to the feeling that their vote has too little impact to matter. In a direct election, the opposite is true, as each vote has precisely the same weight as any other vote.
The electoral college process disproportionately favors smaller states such as Wyoming, which still receives three electoral votes in spite of its small population. In contradiction, California is home to over 70 times the population but only receives 15 times the vote. That means that the vote of Californian has less of an impact than the vote of a Wyoming resident in the final vote tally.