I recently asked this question. There, I was also linked to this other question. Both, however, could benefit from the answer to one fundamental point. What is "aerospace grade"? If sometimes aerospace grade and non-aerospace grade parts are the same exact thing then what makes one aerospace grade and the other not? Why would a company pay millions more dollars for one, but not the other?
If sometimes aerospace grade and non-aerospace grade parts are the same exact thing then what makes one aerospace grade and one not?
Sometimes is the key word here. They happen to be the same, sometimes. The spec for the aerospace grade material might be tighter than is the spec for the non-aerospace grade material that is supposed to be the same as the aerospace grade material. The batch that failed quality control for the aerospace grade material might be regraded as the non-aerospace grade equivalent if that bad batch was within spec for the non-aerospace grade equivalent. The quality control team might well go out to lunch when the manufacturing switches from making an aerospace grade product to the non-aerospace grade equivalent of that product.
By way of analogy, think of the difference between a name brand product you buy in a grocery store and a store brand equivalent. Oftentimes, the producer of the store brand product is the producer of the name brand product. One key difference is quality control. When you buy the store brand product you might get the exact same thing as you would with the name brand. Then again, you might not get the exact same thing with the store brand the next time around. People have the expectation that they will get the exact same thing the next time they buy the name brand product.
Other key differences include documentation, traceability, and accountability. For example, the manufacturing of an aerospace-grade fastener (e.g., a nut and bolt) will have to have traceability to the sources of the materials used to make the fastener. This will need to be documented and the documentation will need signed by a high-level licensed professional engineer (who will lose their accreditation if the traceability turns out to be bogus). The mechanisms and devices used to cut the threads will need to be documented, tested at regular intervals, and signed off by a high-level licensed professional engineer (PE). The manufactured fasteners themselves will have to be sampled and tested at regular intervals, typically to the point destruction. The PE will need to sign off on this as well. The documentation burden is very high for aerospace grade materials, and PEs (who are taught and tested to be extremely ethical) are paid very nice salaries.
Non-aerospace grade materials that are supposedly the same as aerospace grade materials typically have wider specs than the aerospace grade equivalent, and never have the meticulous documentation, traceability, and testing standards that burden the aerospace grade equivalent.
Materials science over centuries has known that materials properties can vary widely with subtle differences in composition, processing, heat treating, etc.
Because aerospace products like aircraft are often designed with very tight margins of safety there is little room for overdesigning parts which adds weight and cost. Each part must be exactly the right combination of stiff and flexible. Every part must be "right" whether it was made this year, 5 years ago or ten years from now. If a part is too strong it might break the part next to it and the aircraft kills everyone. If the part is too weak it breaks, aircraft kills everyone. The part must always be just right.
Using poorly controlled materials to make critical aircraft parts is asking for failure. Period.
Or maybe the professionals are all wrong. Maybe aerospace grade is just because secret magic pixy dust is used.
These days the term aerospace grade "has come to be a fashionable marketing slogan for luxury goods". Originally however, it had it's origins in specifications required for military equipment. No military wants its expensive equipment failing at a crucial moment in a conflict or reconnaissance situation.
Essentially, aerospace grade means that high performance characteristics of materials are required. In one application that could mean a high strength to weight ratio, such a material used for structural parts of a an aircraft or spacecraft. In another application it could mean high strength and low deformation in a high temperature environment such as a jet engine or a rocket motor.
Sometimes there is a trade off, high strength characteristics, but low corrosion resistance, as with aluminium 7075, which is an aluminium alloy containing 5.6-6.1% zinc, 2.1-2.5% magnesium, 1.2-1.6% copper.
It is strong, with strength comparable to many steels, and has good fatigue strength and average machinability, but has less resistance to corrosion than many other aluminum alloys.
In a non corrosive environment such an alloy might be acceptable, but not in a corrosive one.
Generally, aerospace signifies a requirement for a heavier than air "device" to be airborne at some time and potentially under conditions that would place high stresses on many, if not all, components of the "device". For the "device" to perform as required it needs to be both light weight and strong.
The other answers have missed the direct answer...
The Society of Automotive Engineers (SAE) has created a list of standards for all sorts of materials, procedures, processes, etc... related to aerospace. These standards are the Aerospace Material Specification (AMS). And there are a lot of them.
In a sense, it's that simple. Anything "aerospace grade" is something that conforms to an AMS specification.
Something to keep in mind is that there are specifications for some really high quality material, and specifications for cheap low-quality stuff. This idea that "aerospace grade" means high quality is incorrect.