Despite their shared culture and history, why is there conflict between Russia and Ukraine?

There are a few items about your question that need to be addressed. The first question is about Ukraine and Russia's "ongoing conflict." The answer to that is a simple question of sovereignty. In March of 2014, Russia invaded an autonomous region of Ukraine that is known as the Crimea. Traditionally, since independence, Russia has interfered with the politics and government of the Ukraine. They annexed the region of Crimea on March 18, 2014.

The second part that needs to be addressed is the history of Ukraine in the Soviet Union. While Ukraine was one of the original members of the Soviet Union with Russia, that was mostly because of communist party members in Ukraine in the early 20th Century. It is unlikely that Ukraine would have had a choice in the matter anyway because of its strategic location, proximity to Russia, and the strength of the Russian military at that time. The people of Ukraine suffered greatly under Soviet rule, especially in the early period. Russian policy in the 1930's focused on a policy of russification. In 1932 and 1933, millions of people, mostly peasants, in Ukraine starved to death in a politically induced famine (Holodomor) in an effort at ethnic cleansing in Ukraine. It is estimated that 6 to 8 million people died from hunger in the Soviet Union during this period, of whom 4 to 5 million were Ukrainians.

The third part of your question that needs addressed is the notion of culturally similar people and warfare. Just because two countries share a culture in terms of language, religion, or customs does not mean they will not have different economic and political goals that may lead to warfare. The American colonists were not culturally different than the English when they conducted war for independence. Many a civil war has been fought by combatants that are culturally similar. The bottom line is that the Ukraine has been an independent country since the early 1990's, has struggled to become a world capitalist power with some degree of success, and does not want its sovereignty challenged by its neighbors, regardless of shared ethnic similarities.