I've started learning French recently, and I've been having a hard time knowing when to use ça or ca in sentences. What are the differences between these two words and how did they etymologically become different?
Only ça exists as a word in French. What you might have encountered is "Ca", with a capital 'C'. When typing, some people omit the cedilla on capital 'C', but it is considered as a mistake. This post about accents on upper-case letters is also valid for cedillas.
The cedilla modifies the pronunciation of the letter 'c'. Ça is pronounced /sa/; without the cedilla it would be pronounced /kɑ/.
A word pronounced /kɑ/ exists in French, it is spelled cas, it is invariable and takes an 's' even in the singular.
When people do not have French keyboards available, they sometimes write ça as ca. This is becoming much less common as more and more systems are adapted to work with the French alphabet.
The standard spelling is ça. As a learner you should always use ça unless you really have no way to type it.
If you don't have this character on your keyboard, ç, you can almost always write "cela" instead of "ça", although "cela" is more formal than "ça", so you should prefer "ça" in vulgar sentences. The meaning is exactly the same, it's just a matter of style. Ça me semble trop cher = Cela me semble trop cher.