Why do we have both capital and lower-case letters? When did this come about? Are there any languages that use an alphabet system (no ideograms, like Chinese and Japanese, for example), that do not have upper- and lower-case letters?
Arabic does not use capital letters. Korean has a script which might look like Chinese to you, but which is in fact alphabetic, and it does not use capital letters.
Hebrew has no capital letters (though it does have, for five letters, a special version that is used only at the end of a word).
Here’s a useful link: Unicode 14.0 Character Code Charts
Just about every alphabet used in the world is listed in the left-hand column (along with some other symbolic schemes). You can compare alphabets by clicking on their links and opening the associated PDFs.
Interestingly, by my count, it looks like alphabets that use both capitals and lowercase are actually in the distinct minority.
Hebrew does however have a “script” (handwritten) version that is far more different from the “print” version than is the case for the Latin alphabet. Using the “print” version in handwritten documents is considered, well… weird.
True, but English has a script version also and still maintains a capital-lowercase distinction. Hebrew script, similarly to Hebrew print, has no capital letters. Which is what the OP asked about.
From Graphic Design for the Electronic Age, by Jan V. White:
This process apparently began to occur around the time of Charlemagne.