Distrust and mistrust are roughly the same. Both refer to (1) lack of trust, and (2) to regard without trust. But distrust is often based on experience or reliable information, while mistrust is often a general sense of unease toward someone or something. For example, you might distrust the advice of someone who has given you bad tips in the past, and you might mistrust advice from a stranger. This distinction is only a general tendency, though, and it is not always borne out in real-world usage.
Although distrust and mistrust are slightly different, many writers use them interchangeably, usually favoring distrust (which appears about twice as often as mistrust in all English varieties). But the following writers use distrust in its more conventional sense—i.e., in reference to lack of trust based on experience or reliable information:
There’s plenty of reason why this community of ours – a wonderful community in many ways – should distrust the police. [Guardian]
So with the public either distrustful or afraid of the stock market, and market professionals generally looking for a rally failure, what can we expect now? [Forbes]
The profound skepticism shared by many Pakistanis is rooted in their deep distrust of the United States … [Los Angeles Times]
And in the following examples, mistrust refers to general uncertainty or unease not necessarily based on experience, which is its more conventional sense:
Blocking the Internet, meanwhile, can increase fear and mistrust because it cuts off access to accurate information. [Sydney Morning Herald]
They feel victimized, embittered, deeply mistrustful of every established institution except the military. [National Post]