Beyond armament, what major differences separated the galleys used by the Romans and Carthaginians with those used by the Venetians, Byzantines, and Ottomans?

Armament drove almost all the other differences, so it is hard to go 'beyond armament'.

The Renaissance war galley was armed with cannon. The Classical war galley was armed with a ram.

The main differences between classical war galleys and Medieval/Renaissance war galleys were these:

The ram was gone (and so was the above waterline "spur", which replaced the ram in later classical times). Battles were decided by ranged weapons, cannon, and boarding.

The two side rudders had been replaced by one stern rudder.

There were three cannon (usually) mounted at the bow of Renaissance galleys, one 24 or 36 pounder and two 4-12 pounders, plus light swivel guns mounted on the railings along each side.

There was a deck, called the "rambade", over the bow guns which provided a firing platform for ranged weapons.

The lateen rig, usually on two masts, had replaced square rig.

Galleys were heavier built, with frame first, rather than hull first construction. This gave them more protection against cannon.

The rowing arrangement was different, with Renaissance galleys having fewer banks of oars, but more men pulling each oar. Renaissance galleys are thought to have been considerably slower under oars than classical galleys. They did not need to be as fast, since ramming no longer was used as a tactic.

They were probably as fast under sail, and slightly more weatherly.

To determine which design was better, you cannot really discount cannon. The Renaissance galley was designed around cannon.

Cannon were a decisive weapon in boarding actions. Without cannon (or the previous Medieval galley weapon of Greek Fire), classical galleys were quite possibly "better" than Renaissance galleys, but with cannon, Renaissance galleys were "better".