Auto incremented primary keys

If you care deleted/missing key values then Identity is not for you. Check my past answer for more details. Re using an auto incremented primary key[^]

Based on above mentioned reason, Identity is not for display purpose. For internal purpose, Identity is good one.

Identity field is just like Index, not suggestible for searching.

User defined keys are useful when you need non-numeric values like EmployeeID. (Ex. EMP0005). So during search, we could use this value to bring particular Employee's details.

So the answer is Depends.

Start from the basic. What is primary key? It is a key which identify rows uniquely in a table. So we understand that each field of primary key should be unique. It is first condition. To achieve that you can use Identity column or Sequence(sql server 2012). For both type you should declare your field data type int, bigint. You can also use guid as a primary key. If you really need in future your data will go to datawarehouse then consider guid is the first priority. in all that cases database system/application generate unique value for you.
You can also create custom unique value. But the main challenge on that is concurrency handling. So it is better instead of handling concurrency with custom code you can depend on proven system.
So my opinion is you can depend on system for generate primary key.