The volume of the atmosphere is much greater than that of the oceans. However, the mass is much less. The atmosphere masses as much as an ocean covering the entire earth to a depth of 32 feet (approximately 10 M). If the continents were leveled down and thrown into the ocean basins so the earth was one great ocean, it would be several kilometers deep.

As far as which plays a greater role in climate, it is difficult to answer because 1) the two are strongly coupled, and 2) they work in very different ways. The ocean functions as a heat sink with an enormous thermal inertia. It takes a lot to warm it up, and a long time for it to cool off. Heat transfer is mostly by convection. The atmosphere regulates climate primarily by radiative transport and variations in opacity at different wavelengths. If the atmosphere magically disappeared and the ocean could be magically prevented from boiling off and creating a new one, the earth would become frigid very quickly. The oceans would be covered by an insulating layer of ice and then they they would start to very slowly cool off. This happens in the polar regions every winter.

Although the ocean's capacity to absorb heat is enormous, it is not infinite. And you must remember, most of the ocean's water is at great depths and is already only a few degrees above freezing. Most of the thermal interaction with the atmosphere occurs in the upper few hundred meters of the sea.