As a Greek speaker who also knows Latin and Ancient Greek, both words mean the same thing and come from the same roots in their respective language (demos = publius = people/community/population). I don't know why political theorists try so hard to separate them. The only real use of separating them is for easily differentiating the Athenian Democracy from the Roman Republic, for historical purposes, but nowadays both democracies and republics are functionally the same thing (and linguistically should be the same too). The only difference is sometimes the functioning leader's name (president vs prime minister). Every other difference between them are for the sake of local cultural/historical traditions.

In the classical sense, Parliamentary/Representative Republics/Democracies ARE oligarchies. A true democracy would give voting power not just for electing representatives but also for determining specific policies and laws (i.e. Referendums), which very rarely, if at all in many cases, actually happens.