What is the difference between alloy and aluminum wheels?

What is the difference between alloy and aluminum wheels?

There isn't any difference, just two different ways of describing the same thing. The difference will be in how the wheel is formed. 

The best wheels are formed by forging which produces a strong wheel and lets the designer use less material so that it's lighter as well. This strength comes from the excellent grain structure produced from hot forging. These wheels are generally very expensive.

Slightly less strong, and very slightly less expensive than forging is billet machining. With these wheels a billet of material is forged into a shape (round in this case) and the wheel is machined out of that piece of material. The grain structure from this process is not as idealized as the one from forging, so it is not quite as strong. There is also a lot of material loss due to machining. These wheels are almost exclusively multi-piece designs.

More affordable wheels are rolled, which is similar to forging but for the rim only, this adds strength and reduces mass on the outside portion of the wheel which reduces rotating mass. 

Less expensive than that are vacuum cast wheels. These are formed by pouring molten metal into a mold (usually a sand casting) while pulling a vacuum. This pulls any suspended gas out of the molten aluminum and results in a denser wheel with fewer defects, which results in decent strength.

The least expensive wheels are made using gravity casting. The process is similar to vacuum casting above, but doesn't incorporate the vacuum step. This saves a moderate amount of cost, but has a higher chance of yielding a wheel with small amounts of porosity, which can compromise the strength of the wheel. With a good molten metal practice, these wheels can be just as strong as vacuum cast wheels, but at a lower cost.