I have read many articles about Key wrapping and tweakable cipher modes. However, I don't understand the main difference between them? What you recommend for Key storage and Key Archive.
Key wrapping and tweakable encryption are actually quite different topics. Key wrapping is more commonly related to keys forage and key archive.
This is fancy terminology for encryption key with another key. In key wrapping you have a master key that is used to protect one or more keys. It is generally impossible for (computationally constrained adversary) to decrypt wrapped key with no access to appropriate decryption key.
There are key wrapping mechanisms used with asymmetric cryptography, i.e. encryption and decryption with different key (such as RSA-OAEP), and symmetric cryptography, such as AES-KW. Many key wrapping algorithms provide some level of integrity/authenticity protection in addition to encryption.
For purposes of key storage and key archive, key wrapping is very often used.
(As introduction towards tweakable encryption). Key derivation is kind of counterpart of key wrapping. The master key is not used to protect wrapped key, but instead a key derivation algorithm is used to produce other keys from master key. The algorithm may, for instance, generate the keys based on purpose (as identified by "Label") and possibly, other fields related to context. Key derivation can be useful when many keys are needed, but it is not desirable to store all the keys, but only store the master key. (Thus, for key storage, this algorithm is not ideal.)
In some fields, most notably Full Disk Encryption, it is would be desirable to use very large number of cryptographic keys, for instance, to use different cryptographic key per each sector. Conceptually, this is very close to using key derivation to derive per sector keys. Unlike key wrapping, in tweakable encryption the per sector keys are not (usually) stored anywhere.
The tweakable encryption algorithms have been usually implemented with optimization goal that it is very lightweight to change one parameter (tweak). For purpose of key storage and key archive, in many cases, current tweakable encryption algorithms are of little use. Tweakable encryption (for instance XTS-AES), is mainly intended to protect (lower value) bulk data, in situations where data size extension to allow for integrity/authenticity checking is not an option.
Key wrapping is used to... wrap keys :) One example: Suppose you want to encrypt a byte array with a public key algorithm like RSA. Since RSA is slow you would rather choose a random key for a symmetric block algorithm like AES, then encrypt the byte array with AES and just encrypt the AES key with RSA.
Now the question is, how to encrypt the AES key with RSA. The trivial approach is to just convert the AES key into a number (for example by the PKCS#1 standard) and then encrpyt this number with the RSA public key. This has some disadvantages. The biggest disadvantage is, that you have to pad the key, since an AES key is at most 256 bits long while the RSA plain text room is like 1024 bits and above. But padding algorithms are often the source for weaknesses. It is just hard to prove that a specific padding scheme is safe (or at least seems to be safe).
A much smarter way to encrypt the AES key is the following:
To decrypt your message just:
This is an example for a key wrapping algorithm. It's not designed to encrypt predefined large byte arrays. It's designed to encrypt short data. In this particular example it just creates and then wrappes a random byte array.
Tweakable encrpytion is a mode for symmetric block ciphers. For example CBC mode encrypts the first block with your key (and some initial vector). The key to encrypt the second block then depends on your first block and so on. For tweakable encryption you almost use the same key for every block. The only difference is, that you incorporate for example the block number in the key.
This way again every block has its own key, but with tweakable encryption you can parallize the encryption, since the key for block number $n$ does not depend on block number $n-1$.