Metallurgy 101
We usually think of metal as a simple solid material, without any internal structure, like putty. In reality, it's a jumble of crystals, all grown into each other. Usually, the individual crystals are small enough that we don't see them, but other times we do. If you can find a big, cast bronze door handle, look at the back of it, where thousands of hands polish it smooth, and etch it with their acids. You will probably see a splotchy pattern, where each different shaded region is a different crystal, in a pattern resembling frost on a window. The process of formation is much the same. The grain size in a wheel is much, much finer than that.

Casting
The quality of cast wheels varies dramatically, depending on process, and sometimes on variables beyond the control of the manufacturer, such as ambient temperature or even humidity. In general, pressure casting, in which the metal is pumped into the mold, is better than just pouring it in. Castings tend to be porous -- some carburetors actually leaked fuel through the metal, with no crack or visible flaw present. Porosity is bad, because it means there are places where the metal isn't in direct contact with more metal on all sides. Voids, which tend to form in the spaces between crystals (a chicken and egg situation), are where cracks begin. Larger, chunky grains may beget larger voids, and cracks along crystal boundaries will have farther to travel. All these points mean that cast wheels must contain more metal to achieve an acceptable strength, and are thus heavier. Still, cast wheels can be made to a high standard with attention to quality processes. The vast majority of alloy wheels are cast, and provide many years of good service.

Forging
Forged wheels take advantage of what happens when metal is cold worked. Cold working doesn't necessarily mean you'd want to touch the materials while they're in process, it means the procedures are done at a temperature below the point where the metal starts to melt and regrow a new crystal structure. Just as the spaces between a metal's crystals may hold flaws, the crystals themselves are full of imperfections called discontinuities. They may take a variety of forms, but discontinuities all share one important quality. By traveling through the crystal lattice of the individual grains, they allow the metal to change shape without fracturing like a diamond. When a load is applied to a metal object, it deforms slightly. When the load is removed, it regains its original shape. This happens because discontinuities move a little, and move back. If the load is high enough, the discontinuities will move until they reach the edge of their crystal, or until they run into another discontinuity.

Generally, discontinuities move one atom at a time, and their movement is guided by the regular structure of the crystal. If a discontinuity in the structure runs into another, the regularity is interrupted, and they may become tangled, and can't return to their starting position. This has two effects. 1) When the load is removed from the metal, it will not return to its original shape. 2) The metal is more resistant to deformation in the future, because there are fewer discontinuities available to move around. This description of the process is a single case of what is actually happening by the billions.

What we can measure is the average of them all. The idea behind forging is to get, on average, the right number of discontinuities tangled around each other, with crystals oriented in the right direction, so that the metal is very strong and resistant to further deformation. This is a delicate balance, because too much cold working makes the metal brittle, so that it fractures instead of absorbing loads. You can see how this works for yourself: Bend a paper clip back and forth many times until it breaks. It begins soft, then gets stiffer, before finally fracturing.

Forging also changes the shape and alignment of the crystal structure. When molten metal solidifies, its grain structure is non-directional, amorphous, grains in the sense of "grains" of sand. As metal is forged, these grains are stretched in the direction of deformation, making them more like the "grain" of wood. The metal is formed so the grain goes in the directions where strength is needed most. Think of particle board versus real wood. One is cheap, heavy, and easily formed into a variety of shapes. The other is strong and light. The forging process, because of the vast pressures involved, also compacts the metal, eliminating porosity and the voids that can be a source of cracks or corrosion. The result is that less metal is required to achieve a given strength, meaning lighter, stronger wheels can be made.