We've touched on a number of points, but here I'll only develop the part dedicated to the general size of the cortex and that of the areas specifically dedicated to different modalities, sensory, motor and cognitive.
Different species have different cortex sizes. The increase in overall size (if any) may or may not result in the appearance of new cortical fields in addition to pre-existing ones. However, in addition to an increase in overall size, or its maintenance, there are also changes in the relative sizes of the different areas, and changes within the same area. For example, in platypus (platypus platypus), not only is there an increase in S1 (primary somato-sensory area), but also within S1, an increase in the representation of the beak, which occupies almost 90% of the sensory surface. This increase is related to the density of sensory receptors on the beak and associated changes in the animal's behavior.
Other equally remarkable changes are found in bats, whose auditory cortex is in proportion to changes in the cochlea linked to the importance of echolocation in this animal. For good measure, note the size of the hand in primates, or the sensory representations of the larynx, lips or tongue in sapiens. This latter series of modifications is linked to the development of oral structures, and is accompanied by parallel changes, in the same direction, at motor or premotor level. The so-called Brocas areas are motor and premotor areas, and their enlargement in sapiens is linked to the production of speech, specific to our species.