Note that this answer reflects a change from the actual situation of classical societies, and doesn't take a specific cause into account.
Note also that this answer is written from a largely western perspective.
The largest changes would be:
Agriculture
Agriculture requires a large amount of fertile land. In ancient times a large majority of built on land was agriculture. Living on the seas would make this impossible, agriculture requires an enormous amount of effort to till.
There could be some small scale agriculture near the sea, but your scenario might preclude this all together.
Either way, you may find this map helpful:
There are myriad of implications here, the most obvious ones are potential vitamin deficiencies, and the fact that populations can only live on what the coast and seas can naturally provide.
Limited population
This limited population would severely reduce the development of technology. An overly large percentage of this small population would spend their time acquiring food or trading, leaving very few (if any) people to learn, teach, investigate, and invent.
Social
Smaller populations have always been more close knit. It is also likely that every member of the tribes would be essential to the survival of the water tribes. You might find parallels with the early hunter-gatherers, though you will have better initial technology.
It is likely that you will have decentralised, almost anarchistic tribes, ones with few social conventions (think about male-female relationships [women are less likely to be devalued]), and close family ties.
The young
Young people would be inexorably bound to the land, questioning the wisdom of staying on the sea, mixed with the lust for adventure that young people have. Though this might be suppressed through tribal ties and the demands of hard work, if you have any young people at all in a privileged society, they will set out, or try to at least.
Greek and Roman sailors
The Greeks were the best sailors of the world. Captains would likely pass on their trade to their childeren or other successors. Sea travel in ancient times was the most efficient way of travel. The tribes would likely be heavily mobile, and even trade routes between the Americas and western Europe are not unlikely. In this scenario, whomever rules the waves, would rule the world.
Before the advent of the Romans this was the Greeks, but the Romans only came into prominence due to their large population, which is less likely in this scenario.
That said, the Romans are the ones most likely to successfully adapt to new situations that require innovative technology within the scientific limits of the current society.
Romans were adequate sailors, but brilliant architects.
Chinese sailors
The Asian civilisations has less developed nautical technology, but it is likely that this would change rapidly. The large pastures as the east of china, and large amount of islands in Indonesia would ensure that the Eastern region would maintain its high level of development compared to the rest of the world, with a booming trade and a relatively high amount of food. An Asian dominated society would be more likely during the development of this scenario.
In other words, the Chinese Sea would be a very useful base of civilisation in this scenario.
Economics
Here things get interesting.
Ships were always at the frontier of trading, and trade would likely flourish. This would drive different regions to specialise in the development of specific near-coast goods.
Tunisian tribes would trade in Purple dyes, Gallic tribes in hides, far eastern tribes in silk, Indonesian and Mediterranean tribes in fish, etc.
The limited amount of people would make war unlikely, and trade would rapidly flourish worldwide.
Now if you can think of a way to increase the population after those trade routes have established:
Technology
The development of technology depends on: -Need, -Materials, -Availability of scientists, -Trade.
Two of those are in ample supply in your scenario. Unfortunately as I wrote earlier, limited food supplies are a major concern, and hunting and gathering is not an effective way of getting food.
The second problem, materials is also rather difficult. You will need lots of copper and tin for a bronze-age society to prosper and progress. I.e., you need mines.
If these concerns can be overcome, and population growth can be sustained, you will see a rapid advance in technology.
Architecture
The difficulties of the situation would lead to a severe need for the development of on-water architecture. It is likely that the more intelligent people would be chosen to study architecture to build ship, on-water cities, and machines to help in the acquisition of materials from the brief on-land stays.
This can help the development of technology of course, but as with the Roman empire could lead to a stagnation of scientific development outside of these specific requirements.
Cities
Wood is of course more readily available, and it is likely that you would see the developement of large wooden cities on the ocean. Wood can be treated against rot, but the disintegration is still going to be a problem. If there is any available metal, it is likely that -heavier than water- materials on water would be quickly utilised.
If no metal is available, the cities would need to be smaller, and large parts would have to be constantly maintained. Disaster would never be far away, and fire would have course be highly regulated.
The first cities would likely be formed by tying boats together. Romans would have a clear advantage here. The Romans were excellent ship-builders, and highly inventive in naval architecture.
It is also possible that reefs and rocks would be used as a basis for the cities.
Very large wooden structures are extremely difficult to create, and are very prone to simply being torn apart by the sea, so expect different structures tied together, rather than one massive whole.
Deep sea mining
This would be the ultimate aim of your society. Important minerals can be found in significant quantities at the bottom of the ocean. Unfortunately, Deep sea mining is extremely difficult, and the resources would be between 800 – 6000 m below sea level. Locating the resources would be difficult too, without the kind of equipment that relies on the kind of resources that are there.
Of course the highest up resources would be easiest to find, so that would be Cobalt and Platinum.
A note on specialists
Note that the need for specialists (architects, maintenance crew, sailors, captains) would not necessarily lead to stratification of society, don't forget what I wrote earlier about the tribal nature of society.
It is likely that captains of trade and fishing ships, and other educated specialists would be respected elders of the tribes, but not outright civil leaders.
