I think the trait that really distinguishes the better part of humanity from the rest of the animal kingdom is the capacity to love and care. Homo Sapiens means "wise man". How would you say "caring" man in Latin?
I offer homo benignus. The adjective benignus means something like "good-willing" or "kind"; see the linked dictionary entry for details. I think this captures the desired sense of "caring" pretty well. If you disagree, I or someone else might be able to find something better if you can describe which direction to take it to.
The adjectives benevolens and benevolus are also possible. If you want a comparison of these three adjectives, I recommend asking a separate question.
I propose homo diligens. My dictionary (Latin/Spanish) defines diligens as appreciative or committed. I particularly like this word because the verb it comes from, diligo, is defined as love stemming from choice and reflection (as opposed to amo, which is defined as love stemming from spontaneous inclination). So homo diligens would be a creature that loves rationally.
Here are some relevant verbs to choose from, depending on the exact sense of "caring" that you have in mind.
Homo curans means "Caring man" in the sense of actively taking care of people or things because one is concerned about them. The verb cūrō suggests devoting attention to the object of your care, to protect that person or thing from harm and ensure its well-being. Sometimes it suggests worry. It lives on in English in the words "curator", "curate", and "curiosity"—the last from cūriōsus, meaning "tending to take great care or interest in something" or even being anxious about it.
Homo amans, "Loving man", would mean only feeling affection or fondness for someone, not necessarily attending to their needs. The verb amō, "I like, I love", lives on in English "amorous".
The verb dīligō, in Homo diligens, suggested by Carlos Arturo Serrano, means to love someone in the sense of favoring them specially, regarding them highly as compared to others. Quem di diligunt, for example, means "Whom the gods favor." The dī- at the beginning is a shortened dis-, meaning to separate or break apart, as in English "distinguish" and "disperse". The -ligo at the end is a shortened lego, here meaning to choose, which lives in English also in the word "elect". When I look up the present participle diligens, though, I just find "diligent".
You probably already know how the binomial nomenclature for species works, but I'll mention a few things here just in case. The first word (capitalized) is the genus: the kind of living thing that the species is distinguished from. The second word is the differentia: the "specific difference", i.e. what makes it essentially different from other species of the same genus. So, whichever differentia you choose would mean that other human species aren't that way, because that's the distinction that you're drawing. If you call our species, for example, Homo amans, that suggests that, say, Homo neanderthalensis was not loving because loving would be what sets us apart.
The unique word in species names is actually the first word, the genus (always a noun). Many different species can have the same differentia (usually but not necessarily an adjective): for example, Lavandula officinalis (lavender), Rosmarinus officinalis (rosemary), and Salvia officinalis (sage) are all species that stand out because they are used in medicine or cooking (hence typically kept in a monastery's storeroom, an officina).
I figure that you're only planning to choose a phrase for rhetorical effect, to provoke thought, so precise adherence to scientific terminology probably isn't crucial. It helps to know a bit about the customs and connotations, though. And the naming of the genus Homo has an unusual history. When Linnaeus named Homo sapiens, he didn't think that there were any other human species to distinguish us from. So, sapiens actually is meant as the quality that separates humans from all other genera. Also, even the scientific naming of species tends toward the poetic. Homo erectus isn't the only human species to walk upright; it was the first, and that was good enough for its differentia.