Disassemble is to take something apart, like an old car motor, but dissemble is sneaky — it means to hide your true self, like the guy who said he was a mechanic but had never actually seen a motor, much less put one back together.
Disassemble is formed from dis-, meaning reversal, and assemble, to put together. Disassemble, then, is to take something apart, literally or figuratively:
Entering the moon's orbit, the rocket was further disassembled enabling two Apollo Astronauts to explore the lunar surface. (Scientific American)
Lawmakers are, of course, free to disassemble their Frankenbill and pass the measures separately. (Chicago Tribune)
But Mr. Walker and Republican leaders said disassembling unions was not the point at all. (New York Times)
Dissemble, which means to hide one's beliefs or feelings, has a less direct breakdown. (Why would it be direct? It likes to hide!) It comes from the Latin dissimulare, meaning to conceal, from dis- for completely, and simulare meaning pretend. If you completely pretend you believe something other than you really do, you are dissembling:
Pictures have always dissembled — there are millions of snaps of miserable families grinning bravely — but now they directly lie. (Guardian)
Is this not the curse of power, forever compelled to conceal and dissemble? (New York Times)
A sweet religion, indeed, that obliges men to dissemble and tell lies, both to God and man, for the salvation of their souls! (John Locke)
Ditch the dis an it's easy to keep these two straight: (dis)assemble is the opposite of assemble, and (dis)semble is to not resemble yourself.