Well, this is an interesting comparison, not because these words are related, but because they could be mispronounced and therefore, heard interchangeably, although I will start by saying that errant is the word most commonly used, heard and understood, while arrant is more obscure, and may remain that way in the shadow of errant.
So, errant is quite easy, deriving from the past participle for the Latin "iterare" for "to journey", but has a confused etymology from the Latin "errare" for "to wander", which the French adopted for "to travel". I suppose a journey without a destination would merely be wandering, so these are not unjustifiably related, although it does presuppose that the journey is without a destination. Perhaps that a distinction between journey and trip - whether the end is predetermined - but that's a comparison for another day. Therefore, the etymology of errant may be errant itself. His errant writing style made it impossible for the judge to follow his argument. Despite the party being only 6 blocks due south of the interstate exit, her errant driving ensured that she was late. The errant witness was delayed by poor directions. All errant ideas not related to study are driven from students while preparing for final exams. Words with people, things and abstractions. There is also a latent meaning of "mistake" and "wrong", with the implied meaning of guilt, perhaps from the intolerance of people who wander. His errant activities off campus earned him a suspension from college for public indecency. Parents should take care to ensure that their children do not engage in errant behavior. Therefore, we should be aware when describing the activities of people that the context is clear whether mere wandering alone is intended or if there is to be a condemnation about the activity as being outside what is expected or prescribed by society.
Now, arrant has the same etymological roots as errant for wandering (having arguably come from a variant spelling - who says spelling doesn't count?), but couldn't be any further from the prior word. The idea that arrant could describe thorough, complete or notorious, clearly now, the condemnation aspect has taken over to an extreme. She was an arrant fool for investing in the pyramid scheme. Although fool was sufficient, and utter would have been an acceptable modifier, if one were actually necessary, with clearer meaning. He was an arrant snob, disdaining the company of those whom he perceived were his social inferiors. Again, snob is enough. His arrant cowardice prohibited him from seeing even the most tame of horror movies. Just trying to use this word now, it feels as if you would toss the word in with the denigrating term as just a sotto voce throw away. "Arrant idiot," she exclaimed upon learning that he had filed yet another motion for summary judgment after losing the first. So, the word only works in the perjorative sense, as some degree of augmentation on the existing perjorative being applied to someone (fool, snob, cowardice). Personally, I don't have much use for arrant. Because of its aural and literal connection with errant, and its meaning of thorough and notorious, it doesn't lend any additional meaning when used correctly, and any attempted use in sarcasm (arrant politician, arrant leader, arrant bravado) only sounds like errant, or has no effect since the listener wouldn't understand the nuance of the insinuation. Plus, I've got other words to resurrect and bring into common parlance than to work at arrant, although "arrant idiot" has some merit, maybe solely as a catchphrase to describe Plaintiff's counsel!