dissimulate vs dissemble
Solution 1:
They are effectively synonyms. I see them listed as such in each other's definitions.
Both are derived from the Latin dissimulare. (To hide/conceal.)
Dissemble seems to have derived after a trip through old French from a combination of semblance and dissimule.
Dissimulate comes mostly unscathed from Latin.
These derivations are from the Google definitions of both words.
Solution 2:
Dissimulate is to pretend to be different. More active/deliberate.
Dissemble is to fail to seem what you really are. Could be by omission.
Simulo in Latin means to pretend. Semblance in English is to do with appearing or seeming. There is a subtle difference.
Solution 3:
My take is that dissimulate is more concealing by omission - just not saying or avoiding questions on the topic. Dissemble is somewhat more active and comes closer to outright lying.