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.