Which Name Is Correct, exFAT or FAT64?

There is no such thing as FAT64 (at least not at this time), but there is exFAT, which some people refer to as FAT64.

Why do they do this?

The history of the File Allocation Table is quite involved. These days, the most common implementations are FAT32 (though this is increasingly uncommon) and exFAT. FAT32 was a significant improvement over the older FAT file systems, allowing volume sizes up to 2TB (with a sector size of 512 bytes) and 16TB (with a sector size of 64KB). That is still large enough for most installations in 2016.

Unfortunately, the largest file size was one byte less than 4GB, which is pretty small nowadays. exFAT does away with the 4GB limit, allowing files well into the PB range, and similarly with the volume size. It does so using 64-bit length fields.

As FAT32 used 32-bit length fields, exFAT naturally acquired the nickname FAT64. So yes, FAT64 and exFAT are the same thing, but exFAT is the correct name.