To impugn is to attack something as false or questionable. For example, if you disagree that this is the definition of impugn, you might impugn our definition.
To impute is to attribute fault or to credit. For instance, when you blame someone for an act, you impute the act to them. Impute is not always negative, though; for example, you might impute your strong moral fiber to sensible parenting. It can be a tricky word—and, fortunately for most of us, it is generally confined to legal contexts—but it’s simple if you think of it as a synonym of ascribe or attribute.
Impute‘s corresponding noun is imputation. Impugn makes impugnment.
Examples
Impugn
Far be it from me to impugn Shakespeare’s verse, but I could have done with less poetry and more puppetry. [Daily Mail]
Court documents suggest the defense strategy is to impugn the credibility of the arresting officers. [Willows Journal]
They called her a drunk, a prostitute, a pornographer, a liar, mentally unstable — impugning her honor and that of her family. [CNN]
Impute
[B]oredom is both the presiding concern and a state of mind we might impute to the author. [Independent]
This data imputes all taxes (including corporate and excise taxes) to various income cohort. [Atlantic]
Understanding just why reactions were so divergent points to a different logic behind the address to the one imputed from outside. [Foreign Policy]