As far as I know, a Caliphate elects a leader for a lifetime term, rather than having regular elections.
Some caliphates choose their leader by a small group rather than have the masses choose, but some argue that it is a matter of logistics (not being able to quickly get everyone's votes) rather than by design. 
Can a Caliphate, in theory, be secular?
I know that there are many Caliphates in the past, but I'm more interested in the design of the original Rashidun Caliphate, though an explanation of how the system evolved throughout history helps.
And by Republic, I mean the more modern ones, commonly referred to as Democracy, as in many parts of Europe and the United States.
 Dr. Ali Muhammad as-Sallabi and Faisal Shafeeq (2007) The Biography of Abu Bakr as-Siddeeq, Darussalam.
The word "caliph" comes from the Arabic "khalifa", which means "successor [of the Prophet]". The caliph claims a religion-based legitimacy, instead of popular support as in republics. The philosophy is totally different. A caliphate's objective is to have a government based on the Sharia, while a republic seeks to have a government based on popular will.
Successions and elections in historical caliphates
Your question seems to be focused on elections and successions in historical caliphates.
Historically, in most of the so-called "caliphate", the post of Caliph was inherited. That's why you have Umayyad caliphate, Ottoman caliphate, etc. that are named after the dynasty that controls it. The notable exception, as you mentioned, was the first four caliphs (so-called Rashidun Caliphate).
Even during the Rashidun Caliphate, the election process wasn't really designed to represent popular will and is totally different from elections in modern republic. The main criteria was who could better advance Islam, and popular support isn't the main criteria. There wasn't really a vote where the leader were selected by the majority like in a papal election.
For example, Caliph Umar was appointed by Abu Bakr (the preceding caliph) during the latter's last days, based on advise from a group of pious people. There weren't even a proper meeting where the pious people got to vote and the votes get counted. Basically Abu Bakr just summoned them one by one, interview them with questions like "Who do you think should take over", "What you think about Dude X", etc. and the final decision was by Abu Bakr. This is accepted by Islamic historians, for example here: Selection of Umar (mp3 - listen from around 2:00). This is totally different from an election in modern democratic republic.
@ChintaLaura's explanation is excellent - well researched and reasoned analysis of the question you asked. Your question however hints at a deeper interest in the function of government and various alternative mechanisms to fulfill that function. I think Fukayama's Origins of Political Order might provide a useful analytical framework. Loosely summarizing the tail end of that work, he suggests that there are three components of government; Power (the ability to establish and maintain order), Legitimacy (subordination to a rule of law) and Accountability.
Both the Republic and the Caliphate attempt to create and maintain order within their states. The chief difference is how they obtain legitimacy and accountability. The Republic (should) arise from the consent of the governed (legitimacy) and be accountable to the people through elections. I have only a very limited understanding of the Caliphate, but my understanding, bolstered by @ChintaLaura's answer is that the Caliphate relies on Sharia for both legitimacy and accountability. The people are able to compare the Caliphate's performance to the obligations described in Sharia, and thereby hold the Caliph accountable.
Fukayama goes on to examine a wide variety of other issues that affect government (for example the role of institutions), and he explicitly discusses governance in the Ottoman Empire.
The major difference is that a Caliphate, is a type of rule/governance that is based upon religiously based heredity, whereas a Republic, is a type of rule that is based on the election of Representatives and a Head of State-(i.e. Prime Minister or President).
The word, "Caliphate", meant, succession or heirdom, that is to say, who would be the rightful successor or heir to Muhammad and his legacy. Caliphs-(and for that matter Sultans), were never elected by the public, but rather, were heirs-(or successors), to the previous rulers. In a way, the earliest Caliphs-(as well as Sultans), were Theocrats, that is to say governing a Caliphate-(or Sultanate), strictly based on Shariah-(religious and theologically based) law.
However, a Republic, meant, "the public rule", and is essentially, not so dissimilar to the meaning of the word, "Democracy". The public/people's choice of their rulers/leaders happens through a widespread electoral process. In a Republic, the public/people, ultimately, determine the fate of their country through the leaders they choose to represent them on their behalf. And unlike a pure or true Democracy, a Republic places a greater emphasis on representational leadership in terms of a country's national future.
A Republic, historically and currently, has never, nor does it adhere to a set of strict religious laws and edicts. Religion, in a republic, tends to have minimal significance-(if any significance at all), in terms of its influence on the lawmaking process. In the case of the United States, the Establishment Clause officially distances religion from the Public Square; however, such is not the case in a Caliphate-(or Sultanate), whereby religion, is at the center of the lawmaking process.