Earlyest forms of tattooing known goes back 6000 BC, which is long after Neanderthals died out.
Of course, nothing is known for total sure, we might find older remains - but based on what we can currently know - it seems unlikely Neanderthals tattooed. Body paint IS a precursor to tattooing though, so...
Hair dyeing could be plausible enough, since it is in essence the same as body-painting, you apply a pigment directly to your person
I'm not sure if Neanderthals did figurative art, but chronologically, Homo sapiens did invent it while Neanderthals were still around. Neanderthals definitely did abstract art (patterns, arbitrary decorations, and such)
Had a teacher philosophy for a while who wanted to ad "Homo Ludens" somewhere in the evolution of the humanoids, just because of that.
Philosophers 
What could be "legal" (in terms of the ICZN, the organization moderating the naming of zoological entities) would be to add this as a "sub-species", since these have practically no real definition, are are very subjective. It would if so be Homo sapiens ludens, and we would of course count as H. s. ludens as well.
But there would be nothing "binding" about it, because of the subjectivity of it
The subjectivity would depend on the number of finds. And there it does get tricky of course. Behaviour, in the use of artefacts, or in the decoration of artefacts, does not need to be a subjective thing. But, it does require finds. And the older the finds are, the less there is likely to be found.
It is very rule-bound, but in essence
Genus-level naming - is entirely subjective, by rule - and this causes endless debates, precisely because of the subjectivity of it. A genus is the *first* name, in the string of latin names:
Homo, Panthera, Tyrannosaurus, CanisExamples of debate include - wether or not
Australopithecus* warrants a separate genus, or wether it "fits better" with
Homo or
Paranthropus (I think, don't quote me on it) - a better example is the even longer
Lynx debate, wether the animal "deserves" it's own genus:
Lynx, or wether it should be a separate species of
Felis - or even
Panthera.
Species-level naming - is entirely objective, by rule - here genetics are the key. An old-fashioned "rule of thumb" is that a species is defined by the fertility of its offspring. Technically polar bear and brown bears are the same species, because their offspring are indeed fertile -
Ursus arctos, but for now, polar bear remains separated as
Ursus maritimus.
But generally speaking, a species-level name has to be thoroughly specified and proven.
With extinct animals, where we have no genetic evidence, defining species is much more challenging - and on top of that, genus-debates are much more rampant.
Sub-species level naming - is again very much subjective, because we have allready genetically defined the species - that's done, the organisms can successfully interbreed, so any further division will be "racial" if you will, many synonyms are used, such as "morphs" or simply: Sub-species.
A good example of sub-species are the two types of "common crow" in Europe, where the northern one is black-and-grey, and the southern one is entirely-black, but they are the same species, they have the same language - they communicate effortlessly with each others, and they have no prejudice towards each others, as they will mate and frequently produce half-gray-morphs in the mixing-zone between north and south.
But
all of this comes down to physical traits. There is no significant evidence of physical differences in human anatomy from right before or right after the advent of figurative art - just as there's no significant change before or after the use of copper, or before or after taming of the horse. These are super-significant innovations of ours, but have more to do with... the rare new discoveries, observations, inventions, that have spread rapidly, than a very new feature in their anatomy.
To be boringly technical - what your teacher would be much more correct in doing - would be to name a
period or a
culture rather than the organism, and as it is - those cultures are allready named and defined
If you're curious, the oldest named culture is "Oldowan culture" and is pre-human, as in,
Homo erectus and similar species, and concerns mostly very basic stone tools. It begins around 2 million years ago.
A more recognizeable culture is for example "Corded ware culture", and is much more recent (ca 3000 BC) and is defined by certain burial rituals.
*Type famously known as "Lucy"
**"Type" is the very first individual/fossil ever recieving the name - and thus the individual/fossil you must compare all else with.
***
Genus species subspecies <---correct writing, always capital Genus, but lower-case species and subspecies, and always cursive (well, if you want to be correct!
) "T-Rex" is famously incorrect,
T. rex is correct.
