This is a potentially controversial issue, since there is no consensus yet on the evolution of the brain, beyond a very coarse-grained chronology. Broadly speaking, neocortical areas are new, hence the term “neo-cortex”. But among cortical areas, there is still some disagreement about which areas emerged most recently in primates.
Based on what we know about development in the womb, along with structural findings, my labmates, who are neuroanatomists, suggest that the “eulaminate” areas — the ones that have sharply defined layers — may be the most recent, evolutionarily, compared to the “agranular” and “dysgranular” cortices, which have less sharply defined layers. These less sharply defined areas are also labeled as “limbic”.
Under this model, it is possible that the primary visual cortex (V1) is the most evolutionarily recent neocortical area, as it is the ‘most eulaminate’ cortex in the primate brain, with strikingly distinct lamina. This claim is where debate may arise, since many people intuitively assume that V1 regions in all mammal species are evolutionarily homologous, and therefore equally ancient. This may not be true — it can be argued that since the V1 of a rodent is structurally simpler than the V1 of a primate, it may be more closely related to the limbic visual areas of primates. In other words, it may be that evolution has promoted complexity in primary sensory areas, and not just in the prefrontal areas at the “top” of each functional hierarchy. Anyway, for now this a potential source of controversy.
My colleagues published a paper on this theoretical framework earlier this year  .
Here’s a diagram from the paper, showing the medial (inner) surface of a rhesus macaque brain hemisphere:
According to the Structural Model, the oldest cortex-like areas are in the innermost ring, and more recent “developments” amount to adding concentric rings that grow outwards and laterally from this ancient inner circle.
This figure from an earlier paper  shows schematically what the different cortical types look like:
By sheer coincidence, another neuroanatomist and his colleagues published a closely related paper  on the very same day!
Here is a highly schematized diagram from the latter paper, representing the same ring-like structure (but based primarily on non-primate vertebrate data).
(For reasons that are too complex to get into here, the neocortex is shown inside the purple ring above, whereas in the earlier diagram the newer developments are shown as emerging outside the oldest ring. The discrepancy has to do with the topology of a flattened brain compared to a 3D structure — in actual fact the two diagrams are saying the same thing. Topologically, the neocortex seems to balloon outwards and laterally from “inside” the allocortical ring.)
Note: It is worth pointing out that all brain areas in all animals are equally “advanced”, in that they have all been produced under natural selection, which has been operating since life emerged. So this answer refers to the chronology of emergence of areas, rather than the more vague concept of “advancement”.
 Garcia-Cabezas, Miguel Angel, Basilis Zikopoulos, and Helen Barbas. “The Structural Model: a theory linking connections, plasticity, pathology, development and evolution of the cerebral cortex.” Brain Structure and Function 224, no. 3 (2019): 985-1008.
 Barbas, Helen. “General cortical and special prefrontal connections: principles from structure to function.” Annual review of neuroscience 38 (2015): 269-289.
 Puelles, Luis, Antonia Alonso, Elena García‐Calero, and Margaret Martínez‐de‐la‐Torre. “Concentric ring topology of mammalian cortical sectors and relevance for patterning studies.” Journal of Comparative Neurology 527, no. 10 (2019): 1731-1752.
A version of this post originally appeared as an answer on Quora.