Rabbi Asher (the "Rosh") moved from medieval Germany to Spain. I know he describes some different halachic practices that were done in Spain (e.g. linen tzitzis); but does he describe the difference in culture? Did he manage the different foods? Language? Communication styles?
It is historically true that Rabbeinu Asher was from Germany (Ashkenaz) and moved to Spain (Sepharad) in the early 1300s and carried the tradition of the Ba'alei Tosafoth with him. However, the timeline of the Ashkenazi/Sepharadi split is unclear. There are clear lines of separation dating back at least 1,000 years, this much is true. However, the Geonic period didn't truly end until midway through the 11th Century, well after Rabbeinu Gershom, widely regarded as the first Ashkenazi Rishon, had gained prominence and began issuing decrees that were followed only by Ashkenazim and not Sepahradim (including a ban on polygamy).
That is to say that, while there was already a split geographically, the development into two prominent schools of religious thought was only just beginning in the 11th Century, or less than 300 years before Rabbeinu Asher gained prominence, and this may have been largely due to geography more than anything ideological.
The fact that the RaShB"A essentially vouched for him and made him the de facto chief rabbi of Spain kind of demonstrates that at that stage, any split was one that the leaders didn't much like.
One thing I know he wrote about was surprise that Spanish Jews had assumed authority from the government to enforce capital punishment, although they apparently used this authority to prevent capital punishment, which is something he continued to endorse.
He also opposed secular learning, which he felt negatively influenced Torah study, and which was common in Spain.
What I find interesting about the Rosh is that he remained an Ashkenazi-centrist, even in his host country. He started a Yeshibha based on the Ashkenaz model, married his sons, exclusively, to members of his own extended family (although he did marry his daughters to Sephardim, probably students at his Yeshibha..).
Another interesting thing to point out is that the Rosh brough the concept of self-martyrdom from Ashkenaz (of Crusader infamy) to Sefarad; during the anti-Jewish riots in Seville in 1391, his great-grandson Jacob ben Asher II took his own life and that of his family, rather than submit to the cross.
That horrific act elicited both shock and admiration from various quarters of the Sephardic community.
Another interesting thing to observe about the Rosh; you of course already mentioned his description of differences in custom. Yet never does the Rosh say a word about pronunciation. It seems, in fact, that the differences between Sephardic and Ashkenazic (Western European) pronunciation of Hebrew was a later development. How and why that came about is for a different discussion.