What are the advantages of Always Encrypted 2016 vs Encrypted Columns?
The major difference I know is

Normal Column Encryption

The normal Column encryption which was introduced from SQL Server 2005 uses function Encryptbycert to encrypt the columns. It is not as secured as compared to Always Encrypted in terms that with this method data is submitted as clear text in SQL Server and this data can be seen from traces. A DBA, who has admin access to SQL Server, can see the data so this is actually not fully secured.

Always Encrypted

With always encrypted the encryption is done at clients app by API, like  ADO.net,ODBC. Drivers are installed at clients end to do this encryption. This will not allow SQL Server to see text data hence not revealing it to DBA's and one who have admin access on SQL Server. Always Encrypted allows clients to encrypt sensitive data inside client applications and never reveal the encryption keys to the Database Engine.

Quoting from Docs.microsoft


  Always Encrypted makes encryption transparent to applications. An
  Always Encrypted-enabled driver installed on the client computer
  achieves this by automatically encrypting and decrypting sensitive
  data in the client application. The driver encrypts the data in
  sensitive columns before passing the data to the Database Engine, and
  automatically rewrites queries so that the semantics to the
  application are preserved. Similarly, the driver transparently
  decrypts data, stored in encrypted database columns, contained in
  query results.


I hope I am able to give you some insight into both technologies.