Difference between blocks and sectors
With reference to this article, there is a line that reads:
Because there are limits to the number of blocks, or drive addresses, that an operating system can address. By defining a block as several sectors, an OS can work with bigger hard drives without increasing the number of block addresses.
What does it mean? What is meant by "operating system can address"? And the subsequent maths isn't clear either. How can 64*512 be less than 64*4?
Answer
Look at it this way. Every block that's used in your operating system's file system to store data requires a certain amount of metadata to be stored along with the actual file data you're writing. e.g: timestamps (created, modified), filename, ownership/permission bits. For files that span multiple blocks, you also have to store the IDs of each of those blocks and the order they're chained together, etc.
Determining block size in an OS is a case of tradeoffs. Every file must occupy at least one block, even if the file is 0 bytes long, so there's something for the file's metadata to be attached to. Unless you can guarantee that your files will ALWAYS be some multiple of the block size in size (e.g. in a 4k block OS, all files are 4k), there will be a certain amount of wastage for the files that don't exactly fit within that block.
Small block sizes are good when you need to store many small files. On the other hand, more blocks = more metadata, so you end up wasting a chunk of your storage system on overhead, tracking the location of all the files.
On the flip side, large blocks mean less metadata, but also mean greater wastage when you're storing small files. e.g. a 1 byte file stored in a 4k block wastes 3.99k of that block.
Each of those blocks must be given an ID number by the OS, so it can be uniquely identified. An OS which uses an 8 bit ID field can track only 256 blocks, and therefore, by extension, only 256 files. But if each of those blocks is actually 1 megabyte in size, then you can store up to 256 megabytes of data.
The article you link to has a typo/logical flaw: they meant 512 BYTES, not 512k, so 64*512 bytes is smaller than 64*4k, aka 64*4096 bytes. Most hard drives shipped with 512 byte sector/block sizes.
However, as discussed earlier, small blocks mean more metadata. With drive sizes now in the 3+ terabyte range, with 512 byte blocks, you had to have metadata storage for 3TB/512 bytes = 6.44 billion blocks. That's one major waste of space. So now they ship drives with 4k blocks, 8 times larger, so you only need metadata storage for 805 million blocks. The total number of possible files has been cut by a factor of 8, but the reduced amount of metadata means you can actually store a larger amount of useable data.
Incidentally, 6.4 billion blocks is larger than what can be addressed directly by a 32bit system. 2^32 has an upper limit of ~4.2 billion, so older 32bit machines could not use the entirety of a 3TB drive. Hence switching to larger block sizes. 32bit boxes can easily handle 805 million blocks.