Why SHA-512/256 when we already have SHA-384? Both of them are just SHA-512 with different IV's and truncation. What's the point of having 2 of them? (Actually we have 3 of them as SHA-512/224 is the exact same thing.)
The truncated versions of SHA2 are introduced in 2005 and in the Cryptographic hash Workshop, in 2005, Kelsey listed the reasons as;

Interoperability and security reasons
Need drop-in replacement for SHA1 (MD5)
Have unbroken hashes of wrong size

ECDSA/DSA key sizes
File and protocol formats



Though not mentioned by Kelsey or any other NIST document, the truncation also helps to mitigate from the length extension attack of the SHA series that is due to the Merkle-Damgård construction's architecture problem.
Using different IV enables the domain separation that enables us to have two different random oracles *. It makes sure that the truncated versions have different results. With the domain separation  $$\operatorname{SHA-224}(m) \neq \operatorname{SHA-256}(m)|_{224}$$ where $|_{224}$ is the truncation

What's the point of having 2 of them

If you have an already implementation that uses SHA-256 and you want to mitigate from the length extension attack then SHA512/256 is a good choice. Also, it possibly may increase attack strength.

Also, when you need only 256 but hash output, if you use the SHA512/384 you need to trim that yourself. Not a good idea in the case of programming and in the case of security since you are using the same random oracle.

The ECDSA/DSA key sizes: ECDSA-384 signing SHA512/384 is a good choice especially for 64-bit devices since SHA512 is designed for 64-bit.

Someone tried to use SHA512 and has problems with TLS 1.2. if you are adopting ECDSA-384 signing SHA512/384 is a good choice as above.

For SHA512/224 and simply for 224-bit usage is due to the Triple-DES security, $112 \cdot 2 = 224$. There is a very extensive answer for this by Fgrieu SHA-224 Purpose.





*A random oracle is like a kind of hash function and the same concept applies to hash functions.