I'm currently creating a world with a population of about 4.5 billion people and 21st century technology. In fact, the world is strikingly similar to ours. Oh, except the islands that is. The entire world (except the poles due to ice build up) is separated into archipelagos with anywhere between 10-30 islands, biggest islands are no bigger Honshu (biggest of the Japanese islands) with anywhere between 15-100 miles between groups. What would this do to the climate/weather? And would there be a major change in species? (Not talking about aquatic life forms)
A Bit Different
There are zillions of contributing factors to climate (we don't even fully understand our own), so I'm just going to take a stab at it: this is speculation based on limited information and observation of systems here.
Hot & Cold - with the major currents being disrupted by archipelagos (some in shallow waters, others, like Hawaii, the product of a slowly moving hot spot), the temperatures at central lattitudes will be very hot, but also very rainy with so much evapotranspiration. The poles, without a stream pulling warm water northward, will be very cold, though slightly less so in their summers.
Stormy - while island archipelagos can be mountainous and even have different climates on one small island, the continuous disruption of landform allows for storms to accumulate and 'travel' further and longer. Islandia is going to have lots of hurricanes at the equatorial region.
Fossil aquifers - you will want these in your more temperate and colder regions. Without major mountain systems, you don't get your large, collecting rivers for fresh water. Rain will not be as prevalent in the temperate and colder latitudes So let's give these guys some spring water.
The temperate zones are going to be the sweet spot, where you can hopefully have seasons a rich diversity of fishing/whaling as the staple meat, and a lot less tropical storms.
Fauna - you didn't ask, but I suspect, like most small island chains, you will have a great diversity, but no large land animals, such as 'lions and tigers and bears, oh my'. Large land animals need a hella lot of area to cover for food, both herbivore and carnivore (usually). Your small herbivores and rodents (although greatly diverse from each other) will be your land animals, but the coolest thing? Your birds might rule the skies: and you might have very large, and strong birds.
Fish - Good luck with this one: I think even though we have separation on Earth that keeps species apart, I would guess we'd see the same patterns here, although with your population, they will be threatened by overfishing.
Dustin covered the weather nicely. What is the tilt of the planetary axis? That affects the size of the polar regions. If there is no land under the poles and no continent to block block ice flows, you may have a huge sea ice problem affecting navigation. If you do not have much in the way of deep ocean (islands most everywhere means they are usually on the continental shelf), then the ocean's ability to moderate the temperature is limited, so you may experience more swings in temperature. Fewer large oceans also means currents like the Gulf Stream do not exist to carry warm water poleward, so islands in the high latitudes will be very cold.
Your world would have many more species of land animals due to population isolation effects on the gene pool.
Since most islands are small, you might have fewer large plains, so no large herds of big herbivores. Expect mostly smaller animals.
Because of the previous issues with water circulation causing cold in the higher latitudes and the inability of non-flyers to migrate toward the equator in most places, I would expect very few non-flyers in the extreme latitudes except those (like polar bears) that fish and can travel on ice flows.
Our oceans have dead zones due to lack of iron far from the continents. Since you have few places far from land, there may be no dead zones, hence much more marine life.
What about vulcanism? A major source of islands is volcanoes, which causes extreme temperature drops, earthquakes and tidal waves. Unless your world does not have plate tectonics. (Many planets are solid - not broken into plates.)
I am probably going to be abusing the archipelago idea with my handwaviumosis. It's a terrible disease.
Your world is a humid one, my friend. I googled the humidity of Indonesia, one of the most populous nations in the world with 240 million people and it has an average humidity of 70%.
The three large equatorial oceans separating the mass archipelagos breed massive hurricanes and cyclones. On Earth these would usually break on land and pour themselves out, but on this world, the high humidity and the large channels of mainly temperate to warm water in-between islands breeds huge storms at the equator during the summer months.
The problem is not the storms, but the fact that they they are constant. In fact, there is a ring of storm clouds encircling the equator of the planet. As if that wasn't enough, the near constant ring of islands separating the equator from the northern hemisphere stops currents like the Gulf Stream from warming the northern waters.
During the winter, the storms at the equator calm down enough, to a more Earthly pattern, and the Ice sheets in the north expand farther than they would on Earth (though the expanded ice caps would be thinner on average).
The majority of the population is supported in the temperate zone in-between the ice sheets and the equatorial storms. They survive by growing crops similar to seaweed and by importing large fish from farms in the sparsely populated equatorial nations. Some cities have domed-off spaces underwater which they grow cash crops and some of the high-priced, surface-extinct vegetables and fruit they consume (I would say grains, but due to the nature of this world grass would be less inclined to evolve as it evolved to take advantage of large, flat, dry plains).
Of course, weather patterns would vary based on many other features, but this is about what I would expect.