My ancestor, August Frederick Friese (https://www.wikitree.com/wiki/Friese-39), immigrated to USA in the 1850s. I recently discovered an August Friese is also connected by marriage to my husband's tree. My husband's August Friese (https://www.wikitree.com/wiki/Friese-157) immigrated to the USA in the 1860s. I haven't yet found a common ancestor between these August Frieses, and maybe there is no connection, especially if this is a very common name.
There were 2935 persons with the last name Friese in Germany in 2002, concentrated particularly in northern Germany (see http://genwiki.de/Friese_(Familienname)#Geographische_Verteilung). Given that your ancestor came from Erfurt (in Thuringia) and your husband's ancestor apparently from Hannover (the city of Hannover is about 110 miles from Erfurt, though "Hannover" can also refer to a larger territory than just the city), it seems probable that they were not closely related.