Common nomenclature if something is "cast aluminum" you pour molten aluminum into a shape that is very close to the final shape of the part. This usually means that you have a relatively complex shape, so there are challenges with getting the molten aluminum to flow correctly and cool correctly, sometimes difficulties with avoiding contamination of the molten aluminum. The more complex the shape, the more difficult it is. There are all sorts of fancy ways to cast aluminum to control this stuff and avoid those problems, and there are also simple sand castings. Cast parts can be good, but they can be challenging to do well, so it's not uncommon for them to have issues.
Common nomenclature if something is "billet" aluminum it comes from a big hunk of aluminum cast in a very basic shape, usually under very tightly controlled conditions (made easier because you are casting a very basic shape, and because one big machine at one facility is doing huge quantities of the same thing so it can be very specialized. The simple shape, and the ability to closely control conditions while the molten aluminum is pouring and cooling allows very good control of the chemical and physical properties of the "ingot", which is the hunk of aluminum you are initially making.
From there, you can either melt it again to cast something, or, the ingot gets cold formed into some sort of useful or at least convenient shape. They're usually rolled into sheets or extruded. A relatively thick "sheet" (or a relatively thick extrusion) can be cut into blocks which are then commonly refereed to as "billet" aluminum.
The cold working increases the strength, and there is usually also heat treating that is used to manipulate the strength.
Fun fact, aluminum is often designated by a four digit number, then a letter and some more numbers "6061-t651", "7075-t6" The first four letters are specifying the chemical composition (how much aluminum and what percentage of what other alloying elements" The "letter" is the temper, it designates heat treating and cold working which determines the strength of the final product. "t" usually involves heat treating and sometimes additional cold working, "H" is designating strength requirements for work hardening alloys that don't readily respond to heat treating, "O" is annealed.
For all of those reasons, the chemical and material properties of what is commonly referred to as "billet" aluminum is usually very well controlled. That's the stuff people are talking about when they say "machined from billet" or "billet aluminum". That tight control over the properties is what is the advantage of "billet aluminum". You are very unlikely to see porosity in "billet aluminum", for all of the above reasons and more.
So if one wants to argue about semantics, yes everything that is made from aluminum involves pouring molten aluminum into a mold and letting it cool and harden at some point in time. Congratulations, go tell somebody at the bar about how your beer can is cast and watch them marvel at your brilliance. They will be amazed, I'm sure.
To say billet and cast is the same thing because all billets are cast is ignoring a whole bunch of metallurgy, either because somebody is more interested in arguing than providing useful information, or because they don't know. Hard to say which, but either way, not being helpful.
