They both basically mean "to lack trust in," but there's a bit of a nuance.
Mistrust has a stronger connotation of suspicion regarding someone's honesty, integrity, or ulterior motives.
Distrust has a bit less weight on that, and more on the mere unreliability.
You might distrust someone with your nice dishes because you think they're not careful and might drop them, but you would MIStrust them with your nice dishes if you think they're borrowing them with the intent of stealing them.
Besides this nuance, you can almost always substitute one for the other, and 99.9% of people won't even care.
Mistrust means “to doubt, to lack confidence in,” as in I mistrust his ability to persuade her. Distrust means much the same but adds suspicion to the mix: He distrusts her because he thinks she’ll cheat him.
(The Columbia Guide to Standard American English. 1993.)
"Mistrust" suggests vague doubts. "Distrust" is rather more emphatic suggesting positive suspicions and even a complete lack of trust.
Also "mistrust" is the preferred form when expressing doubts about oneself.
After researching this in Wikipedia, several dictionaries, Columbia Guide, etc. these are two tough words to tease apart. There is a lot of explicit contradiction between and even within sources. Both words are defined as "suspicious" and "doubtful". "Suspicious" implies the other party's intention to deceive, "doubtful" implies the other party's lack of ability to perform, but not malintent (Urban Dictionary). Reference.com actually defines "mistrusting" as "to be distrustful". Words formed with "mis-" are often used to indicate poor performance without malintent, eg. "mishandle", "misaddress", "misalign", "misapply", "miscalculate", "misinform". But sometimes it does imply intent, eg. "misappropriate", "mislead", "misbehave". Words formed with "dis-" are more often used in a context to indicate a more detached or formal absence or withholding of something, eg. "discredit", "disallow", "disability", "disuse", "disunity". For example, to discredit a theory is to show it has flaws, but not necessarily to show that its proponent had deceitful intentions. However! "Disinform" implies intention to deceive while "misinform" implies simply incorrect information. At the end of the day, neither is used with complete consistency. It seems to me that "distrust" *as it is used* *more often* implies that trust should be withheld for some reason, often just prudence. We distrust someone on the other side of the negotiating table because we need to watch out for our own interests; but we do not believe they are intentionally deceitful or we would probably walk away. On the other side, I "mistrust" him because he has misled me in the past. Having said that… there is clear disagreement in many authoritative places about the connotations of these words. Both words denote lack of trust. It is not clear which also implies bad intentions. Therefore, if your context does not clarify your connotation, perhaps you should use other or additional words to be more explicit about the basis for your lack of trust. Like name-calling and stuff like that. lol. BTW, the notion of trust in societies is a fascinating subject. Most Americans fail to realize how much our culture is based on trust. It may not be what it used to be, but most Americans are still very trusting of each other. This is not true in many other cultures; or it is true in a very different sense. Our trust is good because it makes our society (personally and professionally) very efficient, it is bad when trust is misplaced (J). Ultimately, words are defined by their usage, not by dictionaries. Dictionaries simply report their usage and *try* to keep that usage from meandering about by specifying it down.