There is very little difference between the two types. In fact, even the core Python developers agreed that there is no real difference and removed IOError in Python 3 (it is now an alias for OSError). See PEP 3151 - Reworking the OS and IO exception hierarchy:

While some of these distinctions can be explained by implementation considerations, they are often not very logical at a higher level. The line separating OSError and IOError, for example, is often blurry. Consider the following:

>>> os.remove("fff")
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
OSError: [Errno 2] No such file or directory: 'fff'
>>> open("fff")
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
IOError: [Errno 2] No such file or directory: 'fff'

Yes, that's two different exception types with the exact same error message.

For your own code, stick to throwing OSError. For existing functions, check the documentation (it should detail what you need to catch), but you can safely catch both:

try:
    # ...
except (IOError, OSError):
    # handle error

Quoting the PEP again:

In fact, it is hard to think of any situation where OSError should be caught but not IOError, or the reverse.

For clarity, and because some error numbers can differ between platforms (see e.g. this comparison), I would recommend using the errno module in order to catch the specific type of OSError in a cross-platform way:

import errno

try:
    # Code that might fail...
except OSError as e:
    if e.errno == errno.ENOMEM:
        # Handle ENOMEM case...
    else:
        raise

You can use the dictionary errno.errorcode to find the name of a specific error code in the errno module:

>>> errno.errorcode[12]
'ENOMEM'

Note that error number 12 specifically seems to be the same across most platforms, but many others differ.

The expression except (OSError, e) never worked in Python, not in the way you think it works. That expresion catches two types of exception; OSError or whatever the global e refers to. Your code breaks when there is no global name e.

The correct expression for Python 3 and Python 2.6 and newer is:

except OSError as e:

Python 2 also supports the syntax:

except OSError, e:

without parenthesis, or:

except (OSError, ValueError), e:

to catch more than one type. The syntax was very confusing, as you yourself discovered here.

The change was added in Python 2.6 and up, see PEP 3110 - Catching Exceptions in Python 3000 and the Exception-handling changes section of the 2.6 What's New document.

As for an exception for errno.EPERM; you didn't import errno, so that is a NameError as well.

The code that deals with opening and reading from the file should be within the try block of the try/except statement like so:

try:
    # ... preceding code...

    OutFile = open(FileName, 'w')
    for Number in range (NumberCount):
        # Call the function(s) for generating random numbers
        # and writing to the output file
        ReturnNumber = RandomNumbers.RandomWriter ()
        OutFile.write (str(ReturnNumber))
        OutFile.write ("\n")

        print ('Writing ', ReturnNumber)

    # Close the file when finished
    OutFile.close()
except: