Why would Russia be a bigger threat than China?
From a geostrategic standpoint, Russia actually is better positioned than China with relation to the United States. China and the U.S. are too interconnected to seriously consider taking each other on at any point. However, the relationship between the U.S. and Russia is less intertwined. The U.S. depends on Russia for certain raw materials, leading in numerous metals markets, but there is not the same complex producer and consumer relationship that exists between the U.S. and China. Additionally, while China continues to open up its markets to foreign companies and privatizes its industries, Russia remains locked in a more protectionist standpoint where state-owned enterprises dominate critical industries and push out non-Russian competition.
The fact that Russia's main trade is in resources presents them as a greater threat because this is purely a relationship of dependency where other nations need Russia for what it provides and Russia tries to maintain, even increase, that dependency. When a trade relationship is centered on specialized goods it is only necessary that the nation maintain its ability to produce such goods independently by encouraging education and training among the general population. As we see with Russia it has to do what it can to deny customers access to the resources of other countries or somehow take over the resources of those countries, because there are fewer barriers to gaining a foothold in such a market. Where this is best exemplified is the pipeline politics raging through the Former Soviet Union and gradually increasing westward in the European sphere.
In seeking to preserve its grip on critical resource markets Russia has moved to buy out the resource companies of other countries such as in Serbia and Ukraine or force them to transport resources through its territory as has long been the case with the nations of Central Asia and the Caucasus. European leaders, wanting to decrease their dependence on a single foreign power for critical strategic materials, work with U.S. officials to by-pass those routes and this threatens to limit Russia's control of the market. It is in this respect that the Russian relationship with Iran is more intertwined than to China's relationship with Iran. For China Iran is just one of many potential suppliers of resources that are outside American control and a possible customer for Chinese goods. It is more of a mercantile relationship with less strategic significance. China's relationship with countries in Africa or other Middle Eastern nations such as Iraq and even Israel are of more importance in China's strategic calculations. With the latter China has more often found itself aligned with the United States or acting in what is relatively a vacuum of foreign influence. Russia, however, needs to keep Iran out of the U.S. orbit to prevent more open Western access to its resources. Behind Russia, Iran has the largest supply of natural gas in the world. Insuring that a government more friendly to Russian interests maintains control of that supply is critical for Moscow.
Another critical difference between the relationships is one of distance. Russia is very close to the U.S. geographically and this includes the increasingly important Arctic Ocean where both countries have competing interests. At the same time Russia borders Europe and is right at the edge of the Middle East. For the U.S. only Japan and the Korean peninsula would be areas with significant American strategic interests that conflict with China's. While the U.S. naturally wishes to maintain its current grips on the world's waterways the areas in question are not the most critical for the United States. China is still at least a decade away from seriously moving beyond the South and East China Seas into the Indian and Pacific Oceans. However, Russia can very quickly establish a major presence in the Mediterranean and already has a significant presence in the North Pacific Ocean. These waters are far more critical and closer to core American interests, as well as the Arctic Ocean that I previously mentioned, and this puts them in more immediate confrontation.
This means there is a greater likelihood of conflict in the immediate future due to proximity, conflicting core interests, and less cause for avoiding war. While conflict with China would threaten American economic conditions through a loss of commerce, conflict with Russia could actually open up new markets for exploitation that were previously closed to them. For China the same threat to its economy exists, while for Russia it would perhaps give them greater freedom to corner critical markets. Prospects for war being higher makes this a more immediate and volatile threat.
Do not think it is simply about resources though, as there is a complex history between these nations that makes the prospect for conflict more likely between the U.S. and Russia and less likely between the U.S. and China. The easiest way to understand it is that the U.S. and Russia always had an equal relationship. For the first century of America's rise the relationship was one of friendship, with this second century being mainly one of rivalry. In contrast, China's relationship with America began as a relationship of subjugation where the U.S., together with other world powers, controlled Chinese markets and frequently intervened directly in its internal affairs. While seemingly counter-intuitive it actually means they have a less confrontational relationship. Consider it the difference between a parent's relationship with his or her child and a relationship between siblings. Since China is a rising power it sees the U.S. attempts to restrain and control it the way a child sees a parent's attempts to meddle in his or life. However, as Russia has no history of being subjugated by the United States the attempts to restrain and control it are seen as an insult in the same way a sibling would see another sibling's attempts to act like a parent.
For the U.S. it is not even really about Russia, which adds to the insult. The U.S. only seeks to edge out Russia out of its own desire to increase its control over territory. Any restraining of Russia is only done in order to safeguard this newly-garnered control. Since Russia sees the U.S. as an equal this disregard for its interests has caused it to lash out. What is worse is that the U.S. even perceived Russia itself as a "new territory" to control before Putin began consolidating Russian control domestically. It leaves very little room for trust between Russia and the United States.
Militarily speaking, while China is a looming threat, it is not yet ready for the stage of reigning superpower. Russia has long been ready for that stage, but has been frequently blocked by the United States. It lacks the kind of warm water port that can give its navy serious projection power, but it also has no natural barriers to its projection ability as China does. Also unlike China, Russia is not lacking in the expertise necessary to be a reigning superpower, but instead is lacking in the necessary financial resources. Its resource-based economy does give it a better position to build its military, but it will always be limited in what it can do without diversifying its economy. This does not prevent them from successfully launching a major military buildup that can directly threaten the United States in a serious manner as the Germans did prior to World War I, but it probably will prevent them from ever being a reigning superpower without building strategic relationships with other countries.
There is where the greatest threat exists. Russia is a rational power and is thus less likely on its own to ever go to the level of conflict, but unlike China it has to rely on other countries to build its strategic profile beyond its borders. Few options favorable to Russia exist on the current geopolitical map and most of them create ripe opportunities for conflict with the United States. China's strongest strategic allies are countries such as Pakistan that, while prone to conflict, are less likely to drag China into major conflagrations with the United States. Far from it, its relationships tend to put them more closely in line with the interests of the United States.
While it is a much more beloved scenario for certain Americans who want the U.S. to remain the ruling empire in perpetuity, the prospect of war with China is lower compared to the prospect of war with Russia. They are less positioned for conflict in the near future and have fewer core interests in conflict with our own.
Of course, none of that actually means we have to stir the pot. It is, after all, our own figurative victorious humping of the corpse of the Soviet Union that has fueled such enmity between our two countries. Us building missile defenses is not the reason for the tension. Were we less insistent on trying to peel away Russia's support in the region and not attempting to bring all its former subjects into a military alliance that was historically targeted at Russia there may be less opposition to us stationing Arleigh Burkes in the Med armed with SM-3s.