What’s the difference between pls_integer and binary_integer?

Historical reasons. They used to be different before 10g:

On 8i and 9i, PLS_INTEGER was noticeably faster than BINARY_INTEGER.

When it comes to declaring and manipulating integers, Oracle offers lots of options, including:

INTEGER – defined in the STANDARD package as a subtype of NUMBER, this datatype is implemented in a completely platform-independent fashion, which means that anything you do with NUMBER or INTEGER variables should work the same regardless of the hardware on which the database is installed.

BINARY_INTEGER – defined in the STANDARD package as a subtype of INTEGER. Variables declared as BINARY_INTEGER can be assigned values between -2³¹+1 .. 2³¹-1, aka -2,147,483,647 to 2,147,483,647. Prior to Oracle9i Database Release 2, BINARY_INTEGER was the only indexing datatype allowed for associative arrays (aka, index-by tables), as in:

  TYPE my_array_t IS TABLE OF VARCHAR2(100) 
  INDEX BY BINARY_INTEGER

PLS_INTEGER – defined in the STANDARD package as a subtype of BINARY_INTEGER. Variables declared as PLS_INTEGER can be assigned values between -2³¹+1 .. 2³¹-1, aka -2,147,483,647 to 2,147,483,647. PLS_INTEGER operations use machine arithmetic, so they are generally faster than NUMBER and INTEGER operations. Also, prior to Oracle Database 10g, they are faster than BINARY_INTEGER. In Oracle Database 10g, however, BINARY_INTEGER and PLS_INTEGER are now identical and can be used interchangeably.