When should I use them, should I use them at all?
Probably never, unless you're writing historical fiction. Archaic and obsolete words are words that are no longer used in contemporary society, so unless you want to specifically emulate olden times, it's best just to leave them alone.
What's the difference between these descriptions?
According to the Standard English section of the M-W preface, archaic words are older, perhaps at least a century out-of-date and used only for a deliberately old-fashioned effect in modern times, while dated words went out of style more recently. Historical words are words that are still used, but only to refer to ancient things. Rare words are words that are slowly leaving the English language.
I don't see obsolete as a usage marker in my edition of M-W, but I would roughly equate it to being between archaic and dated. Obviously, your dictionary's preface should explain how obsolete would be used by their editors.
Also, should I spend time to remember these archaic and obsolete meanings?
Probably not, unless you're reading a lot of historical stuff. You can always look them up in a dictionary if you're only going to encounter them once in a while.