How did the greco-roman era galley evolve over time? From what Ive heard the late medieval galleys were rather different/improved compared to them?

So I’ve written a bit about it here and here.

From what I’ve read, the galley transformed in a number of ways:

Size: galleys tended to get longer and have deeper drafts, which allowed them to hold more cargo (which was important for ensuring that galleys could offset their operating and repair costs) and lots of oarsmen (150-180 on the great galleys of the Venetian Republic).
Decks: galleys shifted from half-decks to full decks, which means you can put more people on the deck to board/repel boarders/launch missiles. Also, galleys tended to add on fore and/or aft-castles, which were very useful for protecting your ranged marines from boarders and giving them the ability to drop missiles on the enemy deck from above. Also, later galleys tended to have higher sides to help deal with very low-in-the-water vessels like the ubiquitious longship.
Sails: galleys tended to acquire more masts and shifted from square to lateen sails, which allowed them to move faster and more flexibly (as it’s much easier to tack against the wind with that setup).

if you wanted to invest in a massive trade navy, is there any advantage of having longships instead of the regular tradeships the westerosi and essosi use?

Unless you’re dealing with a situation where you want to trade up a riverrine network where ships with deeper drafts can’t go (or if you’re trying to set up a smuggling network where you can’t dock in a normal port and need to beach and then ship out in a hurry) no there is no advantage whatsoever.

Longships aren’t that suited to trade – they’re very small vessels with very limited holds so you can’t carry very much, only one sail as opposed to two or three, they’re very low to the water so rough seas run some pretty serious risks of swamping, etc.

By contrast, your galleys, galleases, cogs, carracks, etc. are much better suited to trade: they’re much bigger so that they can hold more per trip (which leads to economies of scale), they have multiple sails in addition to often having more oars so they’ll generally be faster than longships (which people often get backwards), they’re higher out of the water so they can deal with rough seas more easily, etc.

Since you’re an Historian, I quess you’re the right person to ask :) ASOIAF is full of seafaring vocabulary; since I prefered to read all the series in the original language I could never figure out the different sizes and shapes of all the different ships cited in it (although, I reckon that even if I read them translated, I still couldn’t picture them in my mind to tell the truth…) What’s the difference between a galley/galleas/cog/carrack/etc.?

Good question!

A galley is a ship primarily powered by rowing, tends to be rather long and narrow with a relatively shallow draft. Galleys were the dominant seacraft in the Mediterranean from the classical era through to the 16th century, especially in the era before gunpowder weapons where naval combat focused on ramming and boarding.

A galleas is a heavier galley – they were higher on the sides, they were longer than galleys, and they were slower. They also tended to have more masts and thus more sails than galleys, relying less on oarpower (although they had oars) in order to use the saved space for gun-decks, which meant that they could pack a lot more firepower than a galley. In a sense, the galleas is a transitionary ship between the pre-gunpowder era and the gunpowder era of naval combat.

A cog is a sailing ship without oars, that emerged in the 10th century in the Baltic. Cogs are made from oak, have a single mast, and a square sail. They’re small ships designed for ocean-going commerce, not warships.

Carracks are larger sailing ships than cogs, with three or four masts, which were perfected by the Portugese in the 15th century. With more sails than the cog, you can sail a bigger ship faster, which made the carracks excellent for long sea-voyages and long-distance commerce, because their larger holds allowed you to carry more goods and supplies. When you think about the voyages of Columbus or Magellan, you’re thinking about carracks.