In retrospect, one can see Washington's advice as being more than a tad unrealistic. At the time he gave his Farewell Address, American politics was already in the process of becoming deeply partisan. Washington figured that in warning against the dangers of party and faction he could somehow hold back the rising tide of partisanship. If so, he was profoundly mistaken. In any mature political society it's inevitable that people will have different ideas on how to run the country and on which policies should be adopted to serve those ends.
Although the American colonists had, for the most part, put aside their differences in fighting the British, those differences still remained. And in the post-war environment, with the British safely defeated, it was inevitable that those differences would re-emerge and find expression in the formation of political parties.
Much the same could be said of Washington's strictures against forming foreign alliances. His advice has been ignored because America, of its very nature, has throughout its history needed to get involved in world affairs to defend its own interests. In Washington's defense, one could say that neither he nor anyone else at the time could possibly have foreseen how much smaller the world would become, how much more closely integrated through commerce, trade, and developments in technology. But even at the time he made his Farewell Address, the United States did not—and could not—exist in a vacuum. Even then it was imperative for the United States to form alliances to ensure its continued independence and long-term prosperity.