One thesis, advanced by Harley Balzer that it comes down to Russia having been less open to trade and decentralization. He has a 30-something pages long paper draft arguing this point, e.g. "Russian policy has discouraged rather than stimulated local initiative"; there are no border regions in Russia with growth far outrsripping the rest of the country, like in China. Furthermore there is poorer internal trade in Russia. Some of this is however due to the nature of Russia's economy: "Exporting natural resources does not foster dense economic ties". China was also better at foreign trade deals according to Balzer. Russia joined the WTO in 2012; China back in 2001.
In more observational terms, Russia failed to diversify its exports for over a decade, possibly because it fell victim to the so-called "Dutch disease", fairly commonly affecting countries which heavily rely on exports of raw materials, resulting in an appreciation of their currency in real terms, making some other sectors (typically industrial) less competitive. There are some in-depth analyses (an older one by IMF staff) of this "Dutch disease" in Russia; the diagnostic is only tentative.
Off the top of my head, there were probably other reasons why Russia couldn't hope to take China's place: lower population (somewhere around 1/9 of China today) and higher average salary probably would have prevented Russia from becoming factory central for the world even if its policies had been less hamstringing. Not only was the population smaller, but much less of it was employed in agriculture in Russia compared to China (about 15% in Russia vs 53% in China in the mid-1990s), so there wasn't really much of a perspective in Russia of luring masses of peasants to (newly established) cheap sneaker factories [and similar low-tech industries] which have become pretty unprofitable in China nowadays as well, after salaries rose.)
Also, there was probably more reluctance from foreigners to invest in Russia than in China because the potential market was smaller etc. The actual foreign investment numbers support this view (even per capita, although the latter gap was much less drastic):
Another factor pointed out in a (fairly dated now) comparison was presumably the presence of Taiwan and Hong Kong inverstors:
Besides lacking a surplus pool of agricultural labor, Russia’s incipient private sector suffered from a second disadvantage relative to China. New firms are risky, and it can therefore be difficult for them to raise capital. Chinese firms had a tremendous advantage in this regard due to the presence of an overseas network of successful Chinese entrepreneurs, particularly in Hong Kong and Taiwan. These individuals had the resources and the desire to establish a beachhead in the homeland. As soon as China removed restrictions on foreign direct investment, this money came pouring in. Russia, on the other hand, had to go begging to the IMF.
I haven't seen any figures of the Taiwanese and Hong Kong FDIs, so I'm having reservations that the effect was this significant on an economy of China's scale. They might have had more the effect of a path opener, showing other, bigger investors that it was reasonably safe to invest in China.
More recently however, a clear factor was that the Kremlin definitely engaged in too many military adventures. Sure China flexes its muscles in the South China Sea, or with Taiwan occasionally, but this pales in comparison to what Putin did in countries like Georgia, Crimea, Ukraine etc. Some sanctions as a result, soon followed by junk bond status and other perceptions of a high-risk country. To be fair however, this was not the only factor. Depence on energy exports and sanction hit Russia as a double whammy after the financial crisis:
Yearly average growth [of Russia] between 2000 and 2008 was 6.9 percent, a growth rate only challenged by China (10.4 percent) and India (6.7 percent) among the developing and emerging economies (IMF, 2014, Kudrin, Gurvich, 2015). During the later period, 2009–2013, including the financial crisis, Russia's average growth was 1 percent, those of China and India 8.9 and 7.0 percent respectively (IMF 2014). The high growth rate in the first period was due to the economic reforms in the 1990s that led to structural change, productivity increases, integration with Europe and the rise in the oil price from an average of USD 19.60/bbl in the 1990s to almost USD 150/bbl before the dip in 2008. After that oil prices increased again, but growth did not pick up as expected after 2009–2010; instead it declined gradually.
As a final note, Russia is still ahead of China in GDP per capita (approximately $10,000 vs $8,000) as well as in terms of GDP adjusted for purchasing power parity (PPP); the gap is bigger for the latter, about $27,000 vs $17,000. The following graph of (nominal) GDP per capita over time, although a bit stylized, is also indicative of what happened: