While it's certainly true that many Russian historians like to conceive of an unbroken line of Russianness characterizing Kievan Rus and stretching through various successors to the modern Russian Federation, such a reading is at best lazy and at worst dishonest. As Serhii Plokhy discusses at length in his book The Origins of the Slavic Nations, there's nothing really particularly Russian, Ukrainian, or Belarusian about Kievan Rus'. These identities are largely modern constructions that really began to develop in the eighteenth and nineteenth centuries, and so to talk about Ukraine "becoming culturally separate from Russia" presupposes a problematic narrative of a timeless Russian identity from which Ukrainians deviated. Linguistically, at least, we can identify a common Old East Slavic (though classification of the speech of early Novgorod as preserved in the birchbark manuscripts is tricky), characterized by (among other things) relatively faithful orthographic representation of the yers, preservation of the yat and by use of the letter ѧ (originally representing a vowel like that in French vin) for the sequence /ja/, as in землѧ for Modern Russian and Ukrainian земля.

In the thirteenth and especially fourteenth centuries, while Muscovy was consolidating its power over northeastern Rus', the Grand Duchy of Lithuania was taking control over western and southern Rus' (including much of modern Belarus and Ukraine). In short order the Grand Duchy entered personal union with Poland, and Old East Slavic diverged into Old/Middle Russian and Ruthenian. Linguists generally date Ukrainian and Belarusian as literary languages to the early nineteenth century, but it's important to keep in mind that they did not simply arise fully formed from the pre-modern or early-modern carcass of Ruthenian: Ukrainian and Belarusian were codifications of certain parts of a dialect continuum that had been developing for several centuries.

As can be seen by the other posts here, the historiography on the question is complex and changing. Depending on who you ask and what their background/bias on the topic is, you may get several different answers. But I will contribute another version, one that is a little more modern.

After the fall of Kieven Rus, the territory of what is now modern Ukraine was eventually divided into two spheres: one controlled, and later annexed, by Russia; the other by the Polish-Lithuanian Commonwealth, later replaced by Austria when Poland was partitioned. As a result the split Ukrainian territory developed different identities. The eastern, Russian-based one grew a closer identity to Russia, while the west retained a dominant Ukrainian identity. This division lasted a couple centuries, and created a fairly large rift between the two sides (that in part explains the current issues, but we're not getting into that here).

With the collapse of the Russian and Austro-Hungarian Empires at the end of the First World War, changed the design of Ukraine. Between the Treaty of Brest-Litovsk, the declaration of the Second Polish Republic, the October Revolution and Russian Civil War, and the Polish-Soviet War, everything was happening. Several Ukrainian states were declared at this time, varying from communist republics to more moderate socialist states, and based in both Kyiv and the west. This, I would argue, was the first time a modern, nationalist concept of a Ukrainian state was formed (I say this fully aware that the Hetman and Khmelnitsky's state existed, but predate the modern concept of national identity so don't count them here).

Because the division between Ukraine had effectively created two different concepts of what it meant to be Ukrainian, as well as the larger wars going on in and around Ukraine (the Russian Civil War and Polish-Soviet War the most notable), there was no chance for any of these states to survive long-term. The 1921 Treaty of Riga that ended the Polish-Soviet War split Ukraine again between Russia and Poland, with the Ukrainian SSR being formed as a result in the former. After the Second World War the remainder of Ukrainian territory in Poland was transferred to Ukraine, forming (along with Crimea) the modern border we know today.

In short, I'd say that the occupation/conquest of Ukrainian territory by Russia and Poland in the Middle Ages led to a divergence of Ukraine, and that this combined with the turmoil of post-WWI led to the creation of the first ethnic Ukrainian state.