First, the one legitimate connection between folkish and völkisch needs to be acknowledged. They both stem from the respective words for "people" in English and German; folk and volk. Both terms, of course, are related etymologically because English is a Germanic language.
Okay. The one legitimate similarity is out of the way; the words are related linguistically. Now onto the substance.
Prior to World War I, there was a völkisch Party in Germany, which was "the proponent of aggressive annexationism and racist antisemitism at the time of the First World War."* After the war, in the chaos of Weimar politics, "the movement was held together problematically by its sense of German-völkisch mission and superiority. ... it's sense of "völkisch species purity" as the basis of national greatness."*
Indeed, the völkisch movement stepped into the breech after the Munich Beer Hall Putsch, when the NSDAP was outlawed, with the Great-German Volk Community and the National Socialist Freedom Movement of Great Germany, both of which were eventually folded back into the NSDAP once it was reinstated.
In short, the völkisch movement was clearly allied with, and fed into, the National Socialist movement in Germany. It was Antisemitic, espoused German racial superiority, and relied on bogus pseudo-historical ideas of Aryan "root races" and antediluvian super-civilizations.
Modern folkish Asatru, on the other hand, is none of those things.
From its earliest organized form, the Asatru Free Assembly, those elements who wanted to turn an apolitical religion into a vehicle for a racist political agenda, and specifically a Nazi agenda, were told to pound sand:
[This] Nazi-Odinist identification has persisted down to this day, but most of us had either learned to live with it or simply hoped it would go away if we ignored it.
The Asatru Free Assembly announces an end of that tolerance.
We... sympathize with the legitimate frustrations of white men who are concerned for their kind and their culture. These concerns are fully justified. It is a tragedy that these men are driven to radical groups such as the NSWWP [National Socialist White Workers' Party] because there is no well-known, responsible organization for white ethnic awareness and identity."**
One of the modern-day successors to the Asatru Free Assembly, the Asatru Folk Assembly, makes it clear that folkish Asatru does not condone ideas of racial superiority. Indeed, it expressly states exactly the opposite in its Declaration of Purpose:
The belief that spirituality and ancestral heritage are related has nothing to do with notions of superiority. Asatru is not an excuse to look down on, much less to hate, members of any other race. On the contrary, we recognize the uniqueness and the value of all the different pieces that make up the human mosaic. (source)Does that really sound like the policy of a group dedicated to fulfilling the ideology that led to and was espoused by the Nazis? Of course not!
Let's put this one to bed, kids. Just because the words are similar doesn't mean there's anything at all similar in the substance. Modern folkish Asatru has nothing to do with the völkisch movement of the 19th and early 20th centuries. Folkish does not mean racist.
* The Encyclopedia of the Third Reich, Völkisch Movement entry.
** Stephen McNallen, 1978, quoted in Kaplan, Jeffrey; Radical Religion in America, p. 19.