Both these terms come from Alchemy, a field of magic and science that pre-dated chemistry. The philosopher's stone was a substance that could turn base, or common, ordinary metals into precious metals - turning lead into gold, for example. Until chemistry was better understood and the periodic table was developed (by Mendeleyev in 1869), no one knew this wasn't possible, so people kept looking for it. Eventually, these researches were more organized and public, and alchemy became modern chemistry.
The Elixir of Life was similar, a substance that would grant one who drank it (elixirs are usually liquid) immortality. But it was also believed that the Philosophers Stone might grant immortality, so it was an Elixir of Life for some, as well.
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