How compare the rights and conditions of the American slaves to those of Russian serfs?

Based upon the information that I have about the subject, the only difference between slave and serfdom is the word. But the condition was virtually the same. The word serf was basically the proper word for slave. Similar to how indentured servant was proper word for slave.

One of the important - and rarely mentioned - differences between serfs and slaves is that former, at least in Russian Emprie, could carry guns. And as the famous quote states, "The possession of arms is the distinction between a freeman and a slaves".

Of course, I'm not claiming that serfs were actually free. Neither I claim that this epithet can be used to describe a modern Westerner.

There is actually a brief academic article on exactly this question: Hine (1975), "American Slavery and Russian Serfdom: A Preliminary Discussion" (JSTOR). Summary of salient points:

the two groups of bondsmen experienced similar treatment and conditions despite the disparity between the two societies. The chronological development of the two institutions even follows a roughly parallel course.

With the Russian Code of 1649:

the connection of the serfs with the land was no longer as significant as it once had been. [It] was a rigid and cruel set of restrictions that denied the peasantry virtually any legal escape from their status. Increasingly, serfs were sold without regard to their attachment to the land.

Hine notes that there were state-owned and privately-owned serfs, the latter being "somewhat less than half of the total number of serfs". His discussion is mainly directed at the privately-owned serfs and he concedes the state-owned serfs were better off.

Both the Russian serf and the American slave were under the total domination of their respective masters. By the nineteenth century when both institutions were fully developed, neither serfs nor slaves had any civil or legal rights. They could not acquire property nor contract a loan. Marriages were sanctioned by the masters, as was any effort to "hire out." They were chattels subject to whatever whims, idiosyncrasies and duties the masters might desire the bondsmen to carry out. Serfdom and slavery were systems of absolute control. Institutions such as the church not only condoned but actually encouraged serfdom and slavery. Many men of the cloth in both nations were unembarrassed possessors of servile laborers. Immense sets of restrictions were imposed by the state to control the conduct and behavior of the slave and serf populations. In the United States slave codes were enacted in all states that permitted slavery. In Russia the central government issued decrees (like the Code of 1649) proscribing the behavior of the serfs and establishing penalties for misconduct. Moreover, it was a visionary serf or slave who entertained any realistic thoughts of freedom or manumission. The two systems were essentially closed and any serf or slave who legally escaped from that status was truly the exception.

Only in the rarest of instances were masters punished for the mistreatment of slaves or serfs. Scarce indeed was the serf or slave who had been successful in seeing legal action taken against the arbitrary acts of a misguided or ruthless master. Any serf or slave could be sold at the discretion of the owner. Bondsmen of both nations were advertised in newspapers; families were cruelly split as their members were sold at public auction.

Other than race, one key difference between the two institutions was that serfs were obliged to do military service:

Every adult male in Russia, after the time of Peter the Great, owed the state an annual soul tax as well as military service. Yet a practice evolved whereby not the serf paid the tax, but rather his master did. ...

Military service was no pleasant escape from the drudgery of serfdom. The Russian army was regarded as possibly the worst fate that could befall a serf. "What they [serfs] feared infinitely more than the birch or stick was the proprietor's power of giving them or their sons as recruits."

There's a number of superficial differences between slavery in the United States and serfdom in Russia before 1861. Serfdom differs from slavery fundamentally in that serfdom ties workers to the land. US slavery tied an individual to another individual. I believe there is a minor flaw in @DarekWędrychowski's answer in that serfs were not owned per say by nobility. Rather the serfs were tied to the land. When land was sold, the serfs who resided on it were sold with it.

@DarekWędrychowski's answer also implies that serfs were free to pursue an education or somehow improve their condition, but I don't believe this is correct. Although there may not have been any explicit stipulation in the law that barred serfs from an education etc., societal pressures were likely significant enough that no serf ever realized this potential.

Serfs were technically not enslaved. They were not the property of nobles. They could accumulate money and purchase objects, perhaps even accumulate enough to purchase land, essentially buying their own freedom. But to my knowledge I am not aware of any such cases occurring. In fact, land prices were likely kept so prohibitively high that a serf saving money for his entire life would not collect even close to the amount required.

Now I say superficial because despite the aforementioned differences in ideology, for all intents and purposes serfdom essentially resembled slavery. Serfs had literally no civil rights. They could be beat, punished harshly, divided from their family, starved etc.

I don't believe serfs could be killed without consequences. It is my understanding that slaves in the US could be killed.

It's difficult to give a proper answer, as the situation of slaves was different among various states and the same with different parts of Russia. So some particular conditions of Russian serfs can be similar speaking of one state, while can be a difference if to compare it with another state.

In the matters of economy and law there were quite many similarities. Like slaves in USA, serfs were owned by nobility, could be sold and moved from the land they lived, what differentiates them from serfs in other European countries at that time. They could be also physically punished and had no such civil rights as the right to own land, leave the property of landlord, trade, go to schools, get married without permission of their owners. It ended in 1861 with the Emancipation Manifesto, which in some ways made serf's situation better (but many of them ended without any land or possessions).

Speaking of differences, I believe that the most important one would be in economical matters, particularly in the way they served their landlords. Serfs were forced to give to their owners part of their income and production, or to work for a landlord for particular number of days weekly (as a family). But of course it differed in time and in particular places. There were of course two kinds of serfs - those who had land and those who were living at the court of landlord, as servants. The situation of the latter became dramatic after the abolition, as there was no land they could buy (even if they could afford it).

Also I see the difference in the fact, that the law in the United States was written in a way to stigmatize slaves as worse kind of humans. F.e. the slave code rules regarding wearing clothes or self-education. In Russia there were no such rules and if a serf had the opportunity to live in better conditions, the law didn't stop him from that just because he was a serf.

After Franklin suggested that no serf ever realized the potential of improving their conditions and the right to education, I'd like to introduce a story of Aleksandr Nikitenko, who went to school being a serf and later, as a free man, became a professor at St. Petersburg University. He was emancipated by his owner in 1824, at the age of 20.

In 1824, thirty-seven years before serfdom was abolished in the Russian Empire, a serf youth, descendant of Ukrainian Cossacks, won his struggle for freedom. Not many years passed before Aleksandr Nikitenko rose to eminence as a university professor and literary critic. He occupied important posts in the Russian government, served in the censorship department and on government commissions, and was elected to the prestigious Academy of Sciences. And all this Nikitenko accomplished well before 1861, when Tsar Alexander II emancipated the serfs.

As a serf, Nikitenko was allowed to go to primary school, but not accepted to the secondary school because of his low class status.

I'll quote few other fragments from his autobiography "Up From Serfdom: My Childhood and Youth in Russia". In particular from its foreword, written by Peter Kolchin.

Here's the fragment that helps us to solve the matter of serfs being owned by the state, which was touched by jwenting:

Peasants constituted the overwhelming majority of the Russian population — 83 percent on the eve of emancipation — but not all of these were privately held serfs: about half the peasants (although far fewer in the central provinces) belonged to the state or the crown, and traditionally enjoyed greater freedom than those who were privately owned; in very rough terms, state peasants can be thought of as the Russian equivalent of the American South’s free blacks.

Kolchin adds an important thing to that:

Although noble serfholders legally owned everything on their estates (including the peasants), they usually provided their peasants allotments of land, which they held and cultivated communally and came to regard as their own. (In one popular peasant proverb, the peasant tells the nobleman: “We are yours, but the land is ours.”)

Those two fragments make it clear for me that serfs were peasants literary owned by landlords, together with their land. While "free peasants", in contrast to serfs, were peasants owned by the state.

From all differences in the slave and serf system described by Kolchin, I've chosen for you few that are most important, when speaking of serfs' living conditions.

The first one touches the matter of self-organization of the day-to-day life of serfs. That difference comes from two facts - different position of serf and slave owners, and the kinds of serviced provided to landlords by serfs.

By American standards, Russian serf-holding estates were enormous. Nikitenko’s unusually wealthy owners—the Sheremetev family—possessed tens of thousands of serfs scattered across Russia; in 1850, just two Americans owned more than one thousand slaves. Only a tiny fraction of enslaved blacks in the southern United States (2.4 percent) had owners with more than two hundred slaves, whereas the great majority of bound peasants in Russia (80.8 percent) had such owners. Unlike most American slaves, most serfs lived in a world of their own, where their masters were remote figures whom they rarely or never saw.

As Kolchin adds much later, it led a significant number of serfs to live quite acceptable life in comparison to slaves, especially when the landowner didn't have enough power to manage all of his serfs, which could be placed on land in different parts of the country. Here's the quote for you:

Badly outnumbered, nobles usually felt uncomfortable among their serfs and interacted with them as little as possible. Some (such the as Sheremetevs) were absentee owners; indeed, wealthy nobles with extensive landholdings were almost always absentee owners, since even if they lived on one of their country estates they were remote figures to the peasants who lived on their other holdings. Even when they were not absent, most wealthy nobles had little contact with their peasants, with the important exception of their house serfs, and dealt with them primarily through administrative intermediaries that included a hierarchy of managers, stewards, and representatives chosen by the peasants themselves. In practice, therefore, peasants on some estates were able to enjoy a considerable degree of day-to-day self-rule, even while ultimately subject to their owners' authority.

To understand how different services were changing the day-to-day life of serfs, we have to know two main kinds of such services:

barshchina:

There were two major ways in which landholding nobles exploited their serfs economically. Some serfholders imposed labor obligations (barshchina) on their serfs, who were responsible for cultivating both their masters’ land and their “own” (which also, legally, belonged to the masters)

and obrok:

Other serfholders, however, dispensed with barshchina, and instead required their peasants to pay them a stipulated yearly fee (known as obrok), in money, goods, or both. Such peasants were free to cultivate their allotments full-time, engage in handicrafts, or hire themselves out for jobs either in their native villages or—with their owners’ permission—elsewhere.

It's safe to assume that for serfs, obrok usually meant a better life:

One advantage of obrok, from the point of view of absentee owners, was that it required less direct supervision of serfs; this was also, of course, an advantage from the point of view of the serfs.

(...) On a day-to-day basis, most Russian serfs suffered less direct intervention from their owners than did American slaves, and were freer to organize their own lives. Required, unlike most slaves, to provide their own sustenance, they controlled a higher proportion of their time; this was especially true of those on obrok, but even those on barshchina could call half their time their own.

And some landowners obviously didn't have anything against their serfs getting rich:

From the point of view of serfholders, wealthy peasants could be a substantial asset, because they could be required to pay unusually large obrok fees.

And such richness could also mean having their own serfs:

Rich and privileged serfs existed; a few became fabulously wealthy, and were able — if their owners agreed, which they did not always do — to buy their own freedom. The Sheremetev family even allowed fortunate serfs to buy serfs of their own, registering them in the Sheremetevs’ name so as to conform with the law; in 1810, 165 Sheremetev serfs owned 903 others.

What's very important, we've got here a clear confirmation that serfs could buy their own, personal freedom.

Completely another reason for better serf opportunities to improve their own situation were racial differences in origins of slave and serf systems:

Russian serfdom lacked the racial component of American slavery. Whereas in the United States (and the Americas in general) the vast majority of slaveowners descended from Europeans and the vast majority of slaves descended from Africans, most Russian serfholders and serfs shared the same national, religious, and ethnic background.

What were the consequences of such difference?

Exceptional serfs—like Nikitenko—were able to attend school, socialize with prominent men, and live practically as if they were free; in the southern United States, such opportunities for exceptional slaves were virtually inconceivable. There were exceptional slaves whose lives were in some ways as different from those of most enslaved blacks as Nikitenko’s was from those of most peasants, but in white racism they faced a powerful additional barrier that did not burden the serfs.