Disassemble is the easy one. Most English speakers can infer from the negative prefix dis- that disassemble means the opposite of assemble. Dissemble is tricker because it has nothing to do with putting things together or taking them apart. It means to disguise behind a false appearance.
Dissemble may sound like the opposite of assemble, but it actually has different roots. Assemble comes from the Old French verb assembler, which in turn has roots in the Vulgar Latin assimulare, meaning to put together. Dissemble comes from the Old French dissembler, whose root sembler means to appear or to seem. So dissemble essentially means to dis-seem. In most cases, it’s synonymous with disguise.
Examples
In web writing, correct use of dissemble is hard to find. For example, these writers misuse dissemble in place of disassemble:
That concentration will trigger weather events that will dissemble the global economy. [Duluth News Tribune (article now offline)]
Burglars dissemble a wall to break into SugarHill Studios … [CultureMap Houston]
Google “dissemble,” and you will find thousands more like these. It could be that the original sense of dissemble is dying—but it’s not completely gone. Here are a couple of rare examples of dissemble used correctly:
If we asked candidates the really hard questions, and didn’t let them dissemble and evade … we wouldn’t have a problem. [Naperville Sun (article now offline)]
… [T]he word is that these are Chinese actors who have been hired to resemble and dissemble North Korean fans. [CBS News]
And for good measure, here are two examples of disassemble used correctly:
These fees cover the costs of recycling firms that disassemble the devices and sell the salvaged materials to commodities markets . . . [Guardian]
It will sell for about $18,500, disassembled, without windows or furniture. [New York Times]
