I thought that the biggest difference was in the initial values for the algorithm. If anyone knows for certain if these two algorithms are completely different as shown in the documentation above, I would appreciate it.
Yes the two algorithms are completely different.
Each "word" in SHA-256 is a 32 bit number.
Each "word" in SHA-512 is a 64 bit number.
SHA-256 has 64 rounds.
SHA-512 has 80 rounds.
The block size (input "chunk") of SHA-256 is 512 bits.
The block size (input "chunk") of SHA-512 is 1024 bits.
Both use the same basic algorithms (all SHA-2 hashes do) but there is no conversion from the output of to the output of the other.
For example the null hash (hash empty string) is:
The reason the articles suggested using a chopped down version of SHA-512 is because for modern CPU the 64 bit operations are faster. So if someone wanted faster performance but only wanted 256 bit hashes (for database compatibility reasons) an ALTERNATIVE to SHA-256 would be a chopped version of SHA-512.