In the ancient world, what was the difference between ships considered suitable for sailing along the coast or up rivers, versus those considered suitable for crossing the open Mediterranean?
Of course, navigation was also an issue, but here I'm only talking about the ships themselves. How did the largest, most seaworthy ship considered suitable for e.g. sailing up the Nile, compared to the smallest, cheapest ship considered suitable for sailing across to Italy?
This isn't necessarily a historical question but I'll try my best to answer. Boats are designed keeping in mind the environment they will operate in. This is the same today as it was in the ancient world with the difference being that modern technologies and materials have made it easier to tackle the same challenges faced in riverine/maritime navigation.
There are/were some important design differences between river and ocean ships. Perhaps the main one is the size of the ship's draft -- the distance between the waterline and the bottom of the ship. The hull of a ship sailing the Mediterranean would generally have a deeper keel to provide stability in rougher sea conditions. By contrast, the Nile is calmer and much less deep than even the most shallow parts of the open sea so Nile boats necessitate a shallower keel. The Baris Nile boat described in Herodotus' histories and its modern descendant the Naggr are both almost entirely flat-bottomed for that reason. There are other differences based on the place and period in time such as different types of wood used (in Ancient Egypt for example local Sunt or Acacia wood was the main wood used in Nile Boats while ships meant for navigating the sea would include more foreign woods like Cedar from Lebanon -- different woods have different properties in terms of heaviness, malleability etc... and price) or different methods of propulsion. The flow of the Nile and wind directions meant it was necessary to include oarsmen and rowing for parts of the journey rather than relying entirely on sail. Today sails are still used on modern Feluccas but oars have been replaced by motor engines.
To answer the second part of your question, size is primarily related to the function of the ship and not to whether it will be sailing on a river or in the open sea. Larger ships aren't necessarily more "seaworthy", but are built larger in order to for example carry more cargo or passengers. Smaller boats are easier to navigate and cheaper to build (but again being cheaper does not translate to less seaworthy). It's not true that larger ships were considered more suitable for sailing up the Nile while smaller ones were suitable for sailing across the Mediterranean. Most Nile boats are relatively small (like the aforementioned Baris and Naggr) and most sea-going vessels are larger by comparison (Roman Triremes or the Ever Given). Of course, the fact that a very large ship will not be easily manoeuvrable in a narrow river or canal is a part of the decision as well.
You might be interested in checking out the BBC Documentary The Pharaoh Who Conquered the Sea about the Min of the Desert, a modern reconstruction of an Ancient Egyptian ship from Hatsepshut's voyages to the land of Punt through the Red Sea. It's interesting, available on Youtube and goes a bit into river vs sea vessels as well.