Because the Russian Empire had a difficult time developing Siberia. And China was even farther away than Siberia.

Modern Russia is difficult to administer because of how incredibly large it is. St. Petersburg and Vladivostok are almost literally on opposites sides of the Earth. Before the completion of the Trans-Siberian Railway in the early 20th century, getting to the Russian Far East required traveling thousands of miles in roads that were impassable much of the year, on rivers that were frozen all winter, or sailing there on a voyage that took months.

Eastern Russia had very few people settled in it for centuries, partly because of the daunting geography and brutal climate, but also because until its natural resources began to be discovered, there wasn't much reason to go there. Much of the Russian Far East was barely governed or even occupied most of the time, so it wasn't territory that could be easily defended, much less to invade from.

Even to this day, Vladivostok is the only large commercial and naval port on Russia's Pacific coast, and Russia did not secure it until 1860. Although Vladivostok is now the home of Russia's pacific fleet, it wasn't a substantial naval base in the 19th century, so it couldn't provide much logistical help for a major invasion. Even as late as the Russo-Japanese War in 1905, Russia had to sail its fleet from Europe, around the Horn of Africa, and through the Straits of Malacca to even confront the Japanese Navy, because the meager Pacific fleet had already been destroyed easily. (Correction from "kieslowskifan": the fleet wasn't meager or easily destroyed.)

Russia was relatively poor and undeveloped in the 19th century compared to the other European powers of the time, and it faced many border challenges in the west that took priority over eastern expansion. The 1800s saw the Russian Empire fight costly wars against Napoleonic France, Persia, the Ottoman Empire, Finland, and various Caucasian states. It took most of that century to conquer Kazakhstan and other parts of Central Asia.

Surely, Russia would have exploited the Qing dynasty's weakness if they'd been able to, but they were just spread too thin. Even if Russia had industrialized when Britain did, it still would have been a tall order.

From an earlier answer of mine

Well, the Russian state did try to carve out a chunk of the Qing Empire in the late nineteenth century. There was a considerable Russian effort to gain either direct of indirect control over a large swath of Manchuria as well as the Liaodong Peninsula.

Russian expansion into Manchuria and the Amur River basin during the mid-nineteenth century enfolded a number of ideas and imperatives of Russian colonial expansion that acted to reinforce St. Petersburg's interest in the region. Firstly, there was the idea that both the Amur River basin and the surrounding areas to the south were bountiful with natural resources and would serve a Russia's own version of the Mississippi River- a vast waterway in which would allow Russian trade to flourish. Various imperial geographies highlighted this regions' resources and made explicit connections to the ongoing settlement of the US Western frontier. This was a particularly attractive vision of a modernizing empire, especially after Russia's defeat in the Crimean War and within the context of Alexander II's Great Reforms. The Great Reform Era saw the Russian state try to modernize the Empire while preserving elements of autocratic rule. The long historical process of the Siberian fur trade settlement seemed to give a precedent for the successful acquisition and Russification of this territory and added credence to writers arguing Russia had a Manifest Destiny in Northeast China. An empire in the Amur region would be a natural sequel to the earlier fur trading empires and expansion of the seventeenth and eighteenth centuries. There already was a precedent in Central Asia in which semi-autonomous army expansion led to the absorption of the region into the imperial fold. Many of the officer involved in Central Asian expansion such as von Kaufman were also members of various geographic societies that projected a future Asian destiny for Russia by virtue of geography and Russia's European culture. Although the Amur River basin never became an analogous to the Mississippi as the differences in geography are quite great, this vision was very attractive within governmental circles up through the Russo-Japanese War.

Russian interest in this region was not just the byproduct of imperialist projections. The Russian Army operated on the frontiers in the nineteenth century almost as a quasi-independent force for expanding Russia's sphere of influence and control when the opportunity presented itself. A number of the commanders in the Russian Far East such as K. N Gribskii had begun their careers as junior officers in Central Asia in the 1860s and 70s, where the Russian Army had expanded the empire without the explicit permission from St. Petersburg. After the Boxer Rebellion, many of these officers became insistent that Russian interests had to be protected proactively. This meant occupation of Manchuria and attempts to secure the interest of Russian rail concerns. These officers were also alarmed by Japanese interests in Korea and Manchuria, and thus acted at times without formal authorization.

The relative autonomy of the military on the frontiers reflected the highly divided nature of the Russian government over the direction of Russian imperial affairs. This division was especially pronounced after both the Sino-Japanese War and the Boxer Rebellion. Unrest in Manchuria had led to Russian military occupation in the wake of the Boxers. This duration of this occupation posed a quandary for the tsarist government. Members of the Foreign Ministry and the Finance Ministry, like Sergei Witte wanted to use financial power to gain influence within China and return areas in Manchuria to Beijing's control as part of a larger plan of creating an Russo-Chinese alliance under the leadership of the tsar. For Witte, the expansion of Russian railways into Manchuria were a key component of this restoration of the Qing and would stimulate the economic development of both states. Aleksey Kuropatkin, the War Minister, steered a middle course of keeping Russian troops in key strategic areas, but making large parts of Manchuria into an independent vassal state. Admiral Evgenii Alekseev, commander of the Pacific Fleet and chief officer of Russian troops in the Liaodong Peninsula argued that Russia's imperial destiny was hitched to the Far East and advocated a more aggressive policy. Aleeksev increasingly had Nicholas II's ear with regards to imperial policy, but no single faction had gained significant traction. As a consequence, Russian grand strategy within Manchuria remained in a state of limbo prior to the Russo-Japanese War. Russian troops remained within Manchuria as the Russians had already committed significant resources into this area in the form of railways, even though these seldom reaped much benefit to Russia. Imperial policy fluctuated between extremists like Aleeksev who was a firm believer in the Yellow Peril and Russia's manifest destiny in the Far East, and moderates like Witte that saw that extreme Russian imperial claims diplomatically isolated the empire and reaped few rewards. The consequences of divided colonial policy would be a contributing factor to Russia's defeat in 1905 as the extremists like Aleeksev animated the Japanese to take action to prevent a Russian takeover of the region.

Part of the reason why the Russian incursions into China have remained relatively unknown in public memory is precisely because the Japanese ended any Russian pretensions to empire in China at the start of the twentieth century. Other nineteenth-century Western concessions in China either lasted until the Second World War such as the Shanghai international settlement, or ended only in the 1990s when the leases expired. The result was the Russian interlude in the Qing's last decades was more ephemeral and left less of an impact than the Japanese that replaced them.

Sources

Bassin, Mark. Imperial Visions Nationalist Imagination and Geographical Expansion in the Russian Far East, 1840-1865. Cambridge: Cambridge University Press, 1999.

Schimmelpenninck van der Oye, David. Toward the Rising Sun: Russian Ideologies of Empire and the Path to War with Japan. DeKalb, Ill: Northern Illinois University Press, 2001.