My laptop uses an Intel Atom Z3735F processor, which the CPU specs on the Intel website indicate is 64-bit but only supports a maximum of 2 GiB of RAM. (The laptop came with Windows 10 32-bit pre-installed.)
What is the point of having a 64-bit CPU which only supports 2 GiB of RAM? Does a 64-bit processor have any advantage beyond more conveniently supporting more memory?
The reason is for manufacturing. CPUs have been capable of 64-bit processing for about 15 years, that doesn't mean that they are given enough memory to make running it in 64-bit mode worthwhile.
For a company having to manage both a 64 and 32-bit product line would be a hassle, particularly when the 64-bit version can run in either mode. As a result they can simplify their product offering down and offer a "one size fits all" product that does both. For a company like Intel that means they can produce more of less different types of processor and keep their manufacturing overheads lower.
Then it becomes up to whoever is buying the chips as to how much memory they provide, whether the memory can be upgraded and so on.
Yes, a 64-bit operating system with only 2GB of RAM might be a waste, but a 64-bit processor might actually be a bit faster for certain tasks so the trade-off might be worthwhile to someone who cares about the difference.
This is also partly about CPU binning, where CPUs are made identically but have defects that cause certain parts to fail test or are disabled simply to segregate the market. The z3590 for example is a chip based on the same design that is capable of 4GB of RAM.
Functionally the cores might be identical, but one has more parts enabled, can be clocked higher and supports more features. Potentially they could have come from the same manufacturing line on the same day, but been segregated differently when tested.
The result of "binning" is a more efficient production line. You test and separate components and rather than simply discarding defective parts you simply redesignate them with the defective parts disabled.
Binning ties in to the whole "more efficient" production line thing. One manufacturing process, making one thing, and then making use of what comes out of the other side in a more pragmatic fashion.
Having chips that are segregated and yet still "pin compatible" means that a manufacturer can pick and choose what version they populate while still using the same design of motherboard. They could have a 32-bit z3735 version with only 2GB of RAM, and then they could offer a 64-bit version with z3590 with 4GB of RAM with (in theory) the only other change being the (now 64-bit capable) firmware installed on the motherboard.
It may not have worked out that way, but it is a potential option.
For Intel, the primary advantages are selling more processors (higher total profit) and maintaining broader market awareness (mindshare) with constrained risk of less profitable products being bought. This is similar to not providing features such as ECC in certain models not because of manufacturing defects, packaging costs, or even validation costs.
(Software, which has an even higher ratio of non-recurring expenses to incremental costs, can be sold with premium versions which merely enable certain features.)
The cost of developing and manufacturing (and marketing) a specific 32-bit x86 processor for a smaller market would reduce total profit. Selling a 32-bit-only x86 processor would tend to fragment the software market and such a processor would be more difficult to market since much of x86's value comes from software availability and compatibility (x86-64 processors support 32-bit software as well as 64-bit software).
(For OEM-only processors (those available only to sellers of computer systems) Intel can enforce other restrictions such as screen size of laptops/tablets.)
For users, the advantages include a lower price and support for 64-bit OS and application software (both availability/longevity and benefits of x86-64 over 32-bit x86).
Support for 64-bit software provides the advantage of longer potential software lifetime. The user will both have more time during which software will be available for the hardware and more time for hardware to be available for the software. 64-bit software can also depend on the presence of certain features; even when features are back-ported to 32-bit mode, since 32-bit software can be used on older hardware those features cannot be assumed present. Some software also benefits from direct support for 64-bit operations (e.g., adding two 64-bit integers is a single operation) or from a larger virtual address space (e.g., sparse memory allocation and application-level single-level storage, where files can be directly-mapped into the address space). For Windows, the software transition also facilitated a significant change in the ABI; passing function arguments in registers is more efficient/performant.
For x86 the 64-bit ISA also increased the number of architected registers, which increases performance. (ARM likewise increased the number of registers in the transition to a 64-bit ISA.)