They do both mean death, but they also have other meanings as well that don’t perfectly overlap, such as “destroy” or “be destroyed” for 亡. I wouldn’t fret too much over the English translations… my Japanese kanji dictionary lists しぬ as a meaning for both.
亡 is never used alone as a word, while 死 can be used that way.
亡’s origin is as an ideogram (abstract depiction of a concept) of something fading from view.
死 was originally a compound of 歹 (originally a fragment of bone) and 人. So, a rather crude portrayal of death. It changed shape over time and you can’t see the 人 anymore.
Are there other aspects that you wanted to know about?
EDIT: By the way, here’s an incomplete list of more kanji that mean, or can mean, death: 去, 逝, 没, 故