First of all, the physical memory chips are no different between a 32 and 64 bit computer.
The article talks about how the processor can reference (or address) the memory. In all modern architectures memory is byte-addressable, which means that every byte-sized memory cell has a number (its address), which the CPU uses to identify that cell. A simple 32 bit processor has 32 bit registers, and a 32 bit number can range from 0 to 4 billion, so it can address 4 GB of memory. A 64 bit number ranges up to 18 quintillion something, or 16 exabyte.
As gnasher states, actual processors do not necessarily match simple models, and most x86 32 bit processors had extensions to address more than 4 GB of memory. However older models without these extensions can literally not use more than 4 GB of memory. If you would be able to put in more memory, these processors would not be able to see it.
That does not mean that the article is entirely wrong, but to see that you need to distinguish between virtual memory and physical memory. Virtual memory is the memory that one program sees, physical memory is the memory that is physically installed in your system, and that only the operating system can access directly. The addressing extensions for 32 bit processors (PAE) only apply to physical memory. A single 32-bit program can only access 4 GB virtual memory, while a single 64-bit program can access a lot more.
The article also mentions the advantage of 64 bit multi core processors, but that doesn't have anything to do with the 64-bit-ness, multi core 32 bit processors also exist. That 64 bit and multi core were introduced around the same period was because both are ways to make use of the ever rising number of transistors that fit on a single chip, which was becoming more than was useful for a single core.