How compare the rights and conditions of the 19-th century American slaves to those of the 19-th century Russian serfs?
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.
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:
Kolchin adds an important thing to that:
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.
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:
To understand how different services were changing the day-to-day life of serfs, we have to know two main kinds of such services:
It's safe to assume that for serfs, obrok usually meant a better life:
And some landowners obviously didn't have anything against their serfs getting rich:
And such richness could also mean having their own serfs:
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:
What were the consequences of such difference?
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 impunity. It is my understanding that slaves in the US could be killed.