Any Python file is a module, its name being the file's base name without the .py extension. A package is a collection of Python modules: while a module is a single Python file, a package is a directory of Python modules containing an additional __init__.py file, to distinguish a package from a directory that just happens to contain a bunch of Python scripts. Packages can be nested to any depth, provided that the corresponding directories contain their own __init__.py file.

The distinction between module and package seems to hold just at the file system level. When you import a module or a package, the corresponding object created by Python is always of type module. Note, however, when you import a package, only variables/functions/classes in the __init__.py file of that package are directly visible, not sub-packages or modules. As an example, consider the xml package in the Python standard library: its xml directory contains an __init__.py file and four sub-directories; the sub-directory etree contains an __init__.py file and, among others, an ElementTree.py file. See what happens when you try to interactively import package/modules:

>>> import xml
>>> type(xml)
<type 'module'>
>>> xml.etree.ElementTree
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
AttributeError: 'module' object has no attribute 'etree'
>>> import xml.etree
>>> type(xml.etree)
<type 'module'>
>>> xml.etree.ElementTree
Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
AttributeError: 'module' object has no attribute 'ElementTree'
>>> import xml.etree.ElementTree
>>> type(xml.etree.ElementTree)
<type 'module'>
>>> xml.etree.ElementTree.parse
<function parse at 0x00B135B0>

In Python there also are built-in modules, such as sys, that are written in C, but I don't think you meant to consider those in your question.

First, keep in mind that, in its precise definition, a module is an object in the memory of a Python interpreter, often created by reading one or more files from disk. While we may informally call a disk file such as a/b/c.py a "module," it doesn't actually become one until it's combined with information from several other sources (such as sys.path) to create the module object.

(Note, for example, that two modules with different names can be loaded from the same file, depending on sys.path and other settings. This is exactly what happens with python -m my.module followed by an import my.module in the interpreter; there will be two module objects, __main__ and my.module, both created from the same file on disk, my/module.py.)

A package is a module that may have submodules (including subpackages). Not all modules can do this. As an example, create a small module hierarchy:

$ mkdir -p a/b
$ touch a/b/c.py

Ensure that there are no other files under a. Start a Python 3.4 or later interpreter (e.g., with python3 -i) and examine the results of the following statements:

import a
a                ⇒ <module 'a' (namespace)>
a.b              ⇒ AttributeError: module 'a' has no attribute 'b'
import a.b.c
a.b              ⇒ <module 'a.b' (namespace)>
a.b.c            ⇒ <module 'a.b.c' from '/home/cjs/a/b/c.py'>

Modules a and a.b are packages (in fact, a certain kind of package called a "namespace package," though we wont' worry about that here). However, module a.b.c is not a package. We can demonstrate this by adding another file, a/b.py to the directory structure above and starting a fresh interpreter:

import a.b.c
⇒ ImportError: No module named 'a.b.c'; 'a.b' is not a package
import a.b
a                ⇒ <module 'a' (namespace)>
a.__path__       ⇒ _NamespacePath(['/.../a'])
a.b              ⇒ <module 'a.b' from '/home/cjs/tmp/a/b.py'>
a.b.__path__     ⇒ AttributeError: 'module' object has no attribute '__path__'

Python ensures that all parent modules are loaded before a child module is loaded. Above it finds that a/ is a directory, and so creates a namespace package a, and that a/b.py is a Python source file which it loads and uses to create a (non-package) module a.b. At this point you cannot have a module a.b.c because a.b is not a package, and thus cannot have submodules.

You can also see here that the package module a has a __path__ attribute (packages must have this) but the non-package module a.b does not.

From the Python glossary:

It’s important to keep in mind that all packages are modules, but not all modules are packages. Or put another way, packages are just a special kind of module. Specifically, any module that contains a __path__ attribute is considered a package.

Python files with a dash in the name, like my-file.py, cannot be imported with a simple import statement. Code-wise, import my-file is the same as import my - file which will raise an exception. Such files are better characterized as scripts whereas importable files are modules.

The other answers here may still be a bit vague, so I'm posting a hopefully clearer answer. It's important to note that the title of the question is also a bit misleading in the first place, and a better title in my opinion would be: "What is special about package modules compared to regular modules?".

TL;DR - Short Answer:

Packages are modules too, they are however, a special type of them. Special in the sense that 1. they are "directories" and 2. they may contain special files such as __init__.py and __main__.py.

To Better Understand - Longer Answer:

The point is, packages are a special type of modules, so we need to understand modules in general first, and then what's special about the package modules will make sense too. (Notice: I'll sometimes refer to "package modules" in this answer as just "packages", and vice versa)

So let's talk about modules in general first, as it would be less vague / easier to understand. There are basically two things that we do with modules, we either import them in other modules, or execute them directly by Python.

Importing a module has a single obvious goal, accessing what that module has inside.

Executing a module however, usually pursues one of these two goals:

1. That module is a main module and executing it will start our program (or one of its subprograms).
2. We want to try the functionalities of that module in isolation, i.e., without having to import it first.

Let's make more sense of all these through some examples:

Importing modules:

# bar.py

def talk():
    print("bar")

# foo.py

import bar # <-- importing module "bar"

bar.talk() # <-- prints "bar"