Address (Cryptocurrency) VS DID (SSI)

What is the common and what is the difference?

Let's first define what these 2 things are.

A cryptocurrency Address 🧾 is an id to identify the recipient of the transaction.
In most cases is deriving from the public key 🔑 (in the case of bitcoin Address it's a hash from the public key, HASH160).
It's important for that recipient to prove that he is the owner of that address/id, moving forward to be able to send another transfer 💵 from that address/id and get benefits from funds that belong to it (there are proper records 📝 at the blockchain ⛓ how many incomes and outcomes to that id were, so actual balance could be calculated based on that).
Since the owner of the address has the correlated private key, he could prove that he is the owner of the public key, by any signing challenge, and by putting that public key into HASH160 he could prove that an address belongs to him.

In this case, some of you can have a question: “Then why just do not use the public key as address?” And in the general case, it could be done and could work this way, and address its just one more abstraction on top of that.
On a paper check 📃, the beneficiary can be the name of the bank account holder, institutions, corporations. So at the paper check using an abstract name as the recipient of funds, what makes that paper checks flexible as to payment instruments. The same idea adopting by cryptocurrency.

DID is a decentralized id (one of the core parts of the SSI), what used to identify the identity.
In some did methods DID as well as address from cryptocurrency deriving from the public key 🔑 (even in most of such methods it's also hash from it).
And at the other did methods DID derives from the public and private keys.
The important part here — it has a way for the owner of DID to prove that he really the owner, moving forward he could prove he is the owner of all digital assets (VCs) 📄 that belong to it. And using the private key, again, by any signing challenge he could prove that he is the owner of the public key and then DID.
In this case, as well the most simple/dummy did method — it's just using the public key as DID, but if you want to have a little bit more complex structure and have a more flexible instrument — you will want probably to have an option to have several keys belongs to one entity, rotate that keys, have different permissions for different keys, so on. In this case, this dummy approach will not work (to solve this SSI has Did Document, and did resolve function), which is a more complex approach, and core SSI technical approaches based on that.

So as a comparison we could say, that Address and DID use for the same purpose — as id 🆔 (recipient of transaction and owner of entity), and from the technical perspective based on the same cryptography principles (deriving from the public key, or deriving from private and public key). But in the case of DID (SSI), it's done in more generic approaches which allow to use it across various of use cases.