We always require a sentence with every question. A specific example helps us think about your question. Do you have a sentence in which you might use these words? If you don't have a sentence of your own, you might give us a sample sentence that a dictionary gives for one of the words, and ask what the difference would be if you used the other. (if you do that, please remember to include the dictionary's name.)
There was no examples so may I say? : I distrust his sly and malevolent smile.
I mistrust his sly and malevolent smile.
I distrust you much more than you distrust mankind.
I mistrust you much more than you mistrust mankind.
And finally is there a difference : I am not distrustful, I just doubt his sincerity.
I am not of mistrustful nature, I just doubt his sincerity.
I think distrust is more common than mistrust, and for some reason I would mainly use mistrust in relation to myself (I mistrust my understanding, I mistrust my memory), but I can't quite articulate why. Maybe others will be able to be more explicit about this. In your examples I would use distrust.