What is the difference between a presidential system and a democratic republic

"Democratic Republic" is more a descriptive expression used by Countries that come out from a form of dictatorship and are eager to emphasize they are now democracies. 
There's also a lot of misunderstanding about the terms "republic" and "democratic" (for some reasons especially between people from the US, perhaps because of the name of American political parties); they imply a "republic" is where you elect somebody to represent you, while a "democracy" is where the majority just tower above a minority. Usually political statements follow :) 
"Democracy" just means power is held by "people" (the citizens), as opposed to governments where one or few people dictate their rules. 
Actually, the difference those people are talking about is between "Representative Democracy" and "Direct Democracy"; all modern democracies are based on Spartan system, not Athenian one, like most believe, since Athenian government was a "direct democracy". 
A Republic is just the most common kind of "Representative Democracy", but not the only one.