This JSFiddle http://jsfiddle.net/SwBLk/1/ may help explain what's happening:
var MODULE = MODULE || {};
MODULE.Submodule = (function(){
var foo = [],
    bar = [];
var init = function() {
    console.log(MODULE.Submodule.foo === foo);
    foo = ['a','b','c']; // This doesn't set MODULE.Submodule.foo
    console.log(MODULE.Submodule.foo === foo);
    bar.push('a'); // But this works as expected
    console.log('foo: ' + MODULE.Submodule.foo); // foo:
    console.log('bar: ' + MODULE.Submodule.bar); // bar: a
}
return {
    init: init,
    foo: foo,
    bar: bar
}
}());
MODULE.Submodule.init();
The first boolean check return TRUE because both objects reference the same object.
The second boolean check returns FALSE because you have replaced foo with a new object and the references don't point to the same underlying object anymore.
You are replacing the foo array with a reference to a new array when you re-assign a new array to it.
When you execute the IIFE, you assign a reference to a very specific  version of "foo" on the return statement. That is the reference you then access when you call MODULE.Submodule.foo. When you go in and replace foo = [ "a", ... ], you are replacing the object foo, but not replacing the reference to it in the MODULE.Submodule object.
EDIT: How do you get around this issue?
1) Ideally, you don't replace the entire array, but only clear & re-initialize it on init() call:
MODULE.Submodule = (function(){
var foo = [],
    bar = [];
var init = function() {
    // Clear our the array if it already contains data (in case your calling "init" more than once)
    // This method is documented to be the fastest way to clear out an array, referenced here:
    // http://stackoverflow.com/questions/1232040/how-to-empty-an-array-in-javascript
    while(foo.length > 0) {
        foo.pop();
    }
    foo.push('a');
    foo.push('b');
    foo.push('c');
    bar.push('a'); // But this works as expected
    console.log('foo: ' + MODULE.Submodule.foo); // foo:
    console.log('bar: ' + MODULE.Submodule.bar); // bar: a
}
2) Your second option is to use what I'd call "dynamic getters" (working fiddle: http://jsfiddle.net/6zVcP/):
var MODULE = MODULE || {};
MODULE.Submodule = (function(){
var foo = [],
    bar = [];
var init = function() {
    foo = ['a','b','c']; // This doesn't set MODULE.Submodule.foo
    bar.push('a'); // But this works as expected
    console.log('foo: ' + MODULE.Submodule.foo()); // foo:
    console.log('bar: ' + MODULE.Submodule.bar()); // bar: a
}
var modify = function()
{
    foo = [];
    foo.push("test");
}
return {
    init: init,
    modifyArrays: modify,
    foo: function() { return foo; },
    bar: function() { return bar; }
}
}());
MODULE.Submodule.init();
MODULE.Submodule.modifyArrays();
console.log(MODULE.Submodule.foo());
This allows you to do whatever you want to the foo object while the getter will always return the latest reference.