A forest of old trees sequesters more carbon per year than a forest with the same quantity of young trees.
When I first saw this question I thought I knew the answer -- trees grow faster when they're young, therefore they sequester carbon at a higher rate. When I went looking for the data to back this up, I found that this is still a somewhat controversial topic. However, based on some recent and comprehensive research, it seems that older trees actually sequester more carbon than young trees, up until their death.
In a 2014 study titled "Rate of tree carbon accumulation increases continuously with tree size" (pdf here), a group of 38 researchers from public and private institutions around the world
conducted a global analysis in which [they] directly
estimated mass growth rates from repeated measurements of 673,046
trees belonging to 403 tropical, subtropical and temperate tree species,
spanning every forested continent.
They found that
In absolute terms, trees 100 cm in trunk diameter typically add from 10 kg to 200 kg of aboveground dry mass each year (depending on species), averaging 103 kg per year. This is nearly three times the rate for trees of the same species at 50 cm in diameter, and is the mass equivalent to adding an entirely new tree of 10–20 cm in diameter to the forest each year.
Essentially, even though old trees are adding mass at a slower rate, kg per kg, than young trees, they are adding more net mass per year because they are already so much larger.
These lines from the abstract sum up the findings nicely:
Thus, large, old trees do not act simply as senescent carbon reservoirs but actively fix large amounts of carbon compared to smaller trees; at the extreme, a single big tree can add the same amount of carbon to the forest within a year as is contained in an entire mid-sized tree.
What about "mature" trees?
In our experience, species we're familiar with such as dogs, cats, and humans reach a maximum size when they've still got well over half of their lives left (barring dietary issues). Thus we might think the same is true of trees -- after a certain number of years, they "mature," and then no longer increase in mass -- meaning they stop sequestering CO2.
But in primary school we all learned that trees add a new ring each year -- this is how we can tell how old they are. So even when a tree stops getting taller, it keeps adding rings, meaning it's increasing mass throughout the trunk and branches.
In a 2008 study, "Old-growth forests as global carbon sinks" (pdf here), the authors look at the basis for the idea that forests "mature" and become carbon neutral (emphasis added):
The commonly accepted and long-standing view that old-growth forests are carbon neutral [...] was originally based on ten years’ worth of data from a single site. It is supported by the observed decline of stand-level NPP [net primary productivity] with age in plantations, but is not apparent in some ecoregions. Yet a decline in NPP is commonly assumed in ecosystem models. Moreover, it has led to the view that old-growth forests are redundant in the global carbon cycle.
The authors then go on to explain the findings of their study, which included analysis of data from over 500 sites in boreal and temperate forests:
If, however, the hypothesis of carbon neutrality were correct, the
expected probabilities of observing a sink or source would be equal and around one-half, the average sink strength for a random ensemble of forests 200 years old and above would be zero, and the mean CO2 release from heterotrophic respiration would equal the mean CO2 sequestration through NPP [...]. However, we observe this ratio to be well below one on average and not to increase with age. Hence, all three quantitative tests fail to support the hypothesis of carbon neutrality. The currently available data consistently indicate that carbon accumulation continues in forests that are centuries old.
They go on to hypothesize a mechanism supporting their observations:
If old-growth forests reach high above-ground biomass and lose individuals owing to competition or small-scale disturbances, there is generally new recruitment or an abundant second canopy layer waiting in the shade of the upper canopy to take over and maintain productivity.
(The "if" here is important because the research excludes the impact of large-scale disturbances such as fire, insect outbreak, avalanches, etc, as these are not a function of the forest itself.)
Although tree mortality is a relatively rapid event (instantaneous to several years long), decomposition of tree stems can take decades. Therefore, the CO2 release from the decomposition of dead wood adds to the atmospheric carbon pool over decades, whereas natural regeneration or in-growth occurs on a much shorter timescale. Thus, old-growth forest stands with tree losses do not necessarily become carbon sources, as has been observed in even-aged plantations (that is, where trees are all of the same age).
While this study did not look at tropical forests, other studies provide smaller-scale evidence that the same analysis does hold in the tropics.
Young forests may actually be carbon sources
In the same 2008 study, the authors present some evidence that young forests may be carbon sources:
In fact, young forests rather than old-growth forests are very often conspicuous sources of CO2 because the creation of new forests (whether naturally or by humans) frequently follows disturbance to soil and the previous vegetation, resulting in a decomposition rate of coarse woody debris, litter and soil organic matter (measured as heterotrophic respiration) that exceeds the NPP of the regrowth.