Sometimes they are essentially synonyms and it’s okay to just remember them as the same thing but, often there is some kind of nuance difference going on.
In your examples, 次 means “next” as in the next thing in a sequence (used in words like 次回 “next time [in a sequence]”), whereas 来 means more like “yet to come”, so it can mean next as in something that will happen in the future or something that’s approaching (used in words like 来年 “next year” or 未来 “future”). As for 死 and 亡, the main difference between these seems to be that 亡 is used as a bit of a softer substitute for 死. For example, 死ぬ is a very blunt way to say “to die”, whereas 亡くなる is a bit softer and is more similar to the phrase “to pass away” in English. Similar to how we say “deceased” instead of “dead people” in order to show more respect to them. That being said, there is the word 死亡 where both kanji are used so it’s not like their meanings are all that different in the end.
As for more kanji that you come across, I think it’s easier to understand how they’re used differently once you learn more words that use them. That way you can get a feel for the different meanings associated with the kanji. Hope that helps.