The distinction I believe you're referring to is Greco-Roman art from antiquity versus Renaissance-era art that was inspired by and seeks to emulate Greco-Roman art in terms of themes and content. Luckily for you, these are relatively easy to tell apart! The main thing is to focus on the physical medium being used and what formal elements are being emphasized. Lots of this has to do with revolutions within the history of art that peaked in the Renaissance period when technological advancements allowed for art to be created more realistically and more evocatively.
Keep in mind that Greco-Roman art is from Before the Common Era, or BCE, so it is old. When you look at a piece from that period, typically noses or arms have been worn away by time, paint is cracked and peeled, maybe the art itself just a fragment or shard of a bigger work. So, if something looks like it's physically been around for awhile, odds are it is quite old. Also keep in mind most of the art that has survived from that period is in sculptural or architectural; 'paintings' as they hang in a museum today didn't really survive, instead mostly sculptures, frescoes, wall paintings (murals), and mosaics. The figures and subjects typically look more stiff, or anatomically incorrect due to the scientific misunderstandings of the body at the time. Poses were less complex and generally more straightforward. This was a time when one-point perspective had just been discovered, as well, but by no means was it refined. Lighting, also, was not really rendered complexly, so most paintings seem more two-dimensional or "flat".
Renaissance paintings, by contrast, are meant to strive for the ideal represented in Greek art in terms of the idealized body seen through the lens of the Greco-Roman pantheon of Gods, and are typically much more technically sophisticated. While they rely on similar imagery as Greco-Roman art, things are rendered in a completely different fashion. For instance, multiple-point perspective was revolutionized in painting during this period, drawing on the teachings from Greco-Roman art but expanded by scientific invention. Figures typically were more anatomically correct, or at least stylized, with more gesture in sculpture, more definition and shading in painting, and generally more explorations and experimentation with style. Oil painting, in particular, was common in the Renaissance era whereas tempera was de jour in ancient art, particularly in mural-painting; tempera is much more flat, whereas oil gives the impression of being more alive because of how it reflects light. And more generally, lighting is given much more weight in Renaissance art and gives a more realistic impression. Also, typically these works have been much more well-preserved. Statues remain relatively intact and are smoother, paintings have less visible damage, and typically are framed on canvas.