As one door closes an opportunity is forever lost. The effects of man-made extinctions reverberate through time and fill the pool of unreachable truths (Hofstadter).
It's unethical toward the whole human race as well.
For example, Siberia will really like to get rhinos, mammoths and other large animals back, and it is possible to breed them from existing warm-climate ones. And there's a lot of space for them to thrive.
This is an understandable position, but difficult in practice. We're losing "information" in a lot of other ways as well. Habitat destruction is probably the most obvious part of that for many species. It's not enough to have a genome without knowledge of what I'll call that genome's "context of relevance". Which amounts to a vast web of climate, local environmental history, other organisms in that context, and so on. All of that will likewise affect any given organism's epigenetic expression, something else that a naive recording of genetic data will miss.
Likewise, we're destroying what amounts to animal "culture". Go read Elizabeth Marshall's The Tribe of Tiger[1], esp. the part that describes the extremely interesting relationship between the Bushmen tribes of the Kalahari desert and the resident prides of lions.
As far as we can manage, it's still best to preserve as much as possible of our "archives of life" in living form vs. recorded on media somewhere.
[1] http://www.amazon.com/Tribe-Tiger-Elizabeth-Marshall-Thomas/...
Of course, there's much we won't get back. There was no doubt a huge amount of variation in the historical population of the rhinos that just isn't present in the handful that remain.
The biodiversity argument is not a warm and touchy-feely one, but it is certainly an economic and social one that should only enhance the touchy-feely arguments of saving a precious animal. A rhino is a 3 billion year old piece of technology that we were gifted a glimpse of. One that we yet have a lot yet to learn from. And we, as species, smelted down that technology to make a ring or necklace. Imagine if a future civilization a billion years more advanced bestowed their version of the iPhone to us, and we melted it to extract the shiny gold... what fools.
Large animals have a generational gap of, let's say 5-10 years, and have numbers in the tens of thousands. Bacterial generational gaps measure in minutes to hours, and their numbers are so large there are no confident approximations. Unsuccessful bacteria species number in the thousands of trillions. One can only imagine what the numbers are for successful species.
This means that the genetic algorithm running a pathetitc bacterial species has a clock speed something like 100 terahertz. It does 100 trillion calculations per second. The genetic algorithm that runs "on" the human race has a clock speed of around 5 calculations per second. For most animal and plant species, the speed would be significantly less than 1Hz. Guess which of these two makes the discoveries ... (the theory being that viruses, on rare occasions, transplant DNA "discoveries" from bacteria into other species, so while humans do evolve, they rarely, if ever, successfully mutate. Rather we are the product of natural genetic manipulation by viruses and natural selection. The same applies for plants and other animals)
The reason large animals survive at all is that their biomass just does not register on the "success" charts for bacteria. Killing every single plant and animal on the planet and fully converting all that mass into new bacterial biomass would not represent 0.01% change in population size for bacteria. That is the only reason why animals survive : we're simply neither a threat, nor a worthwhile target.