Q. Why does it matter whether you are a Litvak or a Polak?
In ancient times the Ephraimites were slaughtered by the Giladites who found that they said sibboleth instead of shibboleth. Yiddish-speaking Jews were often differentiated by whether they said vas, vos or vus when they wanted to say “what”.
Like the Ephraimites, some Lithuanian Jews could not say a sh, so it was Good Sabbes instead of Good Shabbes, and to this day many South African Jews betray their Lithuanian origins by saying, before making Motzi, “Time to vas“, i.e. to wash their hands.
There were and are different culinary traditions, different ways of praying, even different ways of studying the Talmud. The various groups are also associated with distinctive personal habits and ways of thinking.
The perception is that the Yekke (the German Jew) is precise and efficient. The Litvak (Lithuanian) is intellectual and practical. The Polak (the Polish Jew) is passionate and argumentative.
Whether these perceptions have any real basis is a matter of debate. The debates persist even among Litvaks who have never been to Lithuania and Polaks who have never been to Poland.
There are also debates between Chassidic groups, and at times they actually come to blows, but that is another story.