Well, assuming that the dividing line runs through the Atlantic and Pacific Oceans (let's assume 23 W longitude and 157 E longitude), the oceans of the world will drain in a matter of days. Both the Pacific and Atlantic Oceans (all 4) will drain immediately to a depth of at least 15,000 feet, the Arctic Ocean will drain completely, and the Indian Ocean will drain around Antarctica to the same depth, although that will take a few days longer. Not that the earth has a few weeks to watch the process.
Oh yes, and the atmosphere will also drain off, just like the oceans.
EDIT - And it occurs to me that this needs a bit of expansion. Let's assume that the two halves, once split, are alienly moved apart a few thousand miles in the wink of an eye. Then the entire edge around the cut plane will experience what you might call "horizontal explosive decompression", which will actually become a high wind, as the air farther away from the edge expands/is pulled to the edge and beyond. Once over the edge, the air will fall toward the core, and form a bubble sitting on the cut plane. The edge of the disturbance zone will propagate inwards at the speed of sound, so it will be about 10 hours before it reaches the center of the hemisphere. Assuming the atmosphere is uniformly distributed over the new surface, the average density of the new atmosphere will be 2/3 that of the original. However, the air will not be uniformly distributed, since air will fall off the edge. Frankly, I'm not going to do the calculations to figure out what the new distribution will be, but I suspect that quite a large fraction of the available air will be lost to the old surface. In addition, what air there is will be less dense than a similar amount of air pre-split. The reason is that the surface gravity on the old surface will be less than previously, so the atmosphere will not be as concentrated at the surface. At a guess, the air density will be around 1/3 the present value, or about what you find at the top of Everest. It will take a minimum of a day to reach equilibrium (speed of sound from edge to center of old surface to start redistribution, and more than that for the air to flow to its final location) At the edge, the initial winds will be an appreciable fraction of the speed of sound, since the air is expanding freely into a vacuum, and this will help empty the oceans quickly.
The core of the Earth starts about 1800 miles from the center, so the liquid core has a diameter of ~3600 miles. At the outer limit of the core, the viscosity of the molten iron is roughly 100 poise, or somewhere between water and motor oil. Pressures at this depth are ~300 GPa, or 300 million atmospheres. Viscosity increases with pressure, which increases with depth, so the inner core becomes solid.
With the bisection of the earth, the pressure in the core will drop essentially to zero, since core material can squirt out sideways, and the entire core will become liquid, with an initial temperature in the 4000 to 6000 K range. With the central core liquid, there will be no possibility of the edges maintaining shape, and the earth will immediately begin to flow downwards all around the new flat face. The process will continue until the Earth has reached a sphere with a diameter of about 6400 miles. There will be precious little of the original surface left. What there is will be molten, since the gravitational energy released in the collapse will have nowhere to go except as heat. Well, there will be a certain amount of ejecta normal to the plane of bisection.
The energy delivered to crustal material can be roughly estimated as follows. A mass at the old surface (4000 miles from center) will fall 800 miles (1280 km) to reach the new surface. The new surface will have a gravitational acceleration of 9.8 / 2 ^ 1/3, or about 7.8 m/sec^2. The energy imparted to a 1 kg mass will then be mgh, or just about 10,000 kJ. The specific heat of granite is .8 kJ/kg-K, so the material will heat up about 12,000 degrees. Toss in a correction factor to allow for the decreased gravity at altitude above the new surface, and 9,000 to 10,000 degrees seems reasonable. In other words, the new sphere will be molten where it isn't gaseous.
I've not included the energy required to separate the two halves, and this will mitigate the thermal effects some, but it still ought to be quite a show.