Declaring private variables in header file vs declaring variables in class extension

What's the difference between declaring a @private ivar in the header file and declaring the same ivar in the class extension without @private?

There are a few differences. In short, variables declared in the header file are visible to subclasses and class categories. Variables declared in the implementation are not.

1) Instance variables declared in a class's main @interface block are available to external class categories or extensions, even if those variables are declared @private. E.g.:

// YourClass.h
@interface YourClass : NSObject {
    @private
    int _yourPrivateIVar; 
}
@end

// MyExtension.m
@implementation YourClass(MyExtension)
- (void)reset { _yourPrivateIVar = 0; } // This is allowed.
@end

Instance variables declared in the implementation are not available to external class categories.

2) A base class and its subclass cannot both declare the same ivar in their @interface, even if both ivars are @private. E.g., this is not allowed:

@interface Base : NSObject
{
    @private
    int _foo;
}
@end

@interface Subclass : Base
{
    @private
    int _foo; // Error: Duplicate member _foo
}
@end

If both ivars are declared in a class extension or implementation block then not only does it compile but it works as expected: both classes have their own separate _foo ivars that do not conflict with one another. On other words, both variables are truly private and separate:

@implementation Base {
    int _foo;
}
@end

@implementation Subclass {
    int _foo;
}
- (void)reset { _foo = 123; } // Does not affect base class's _foo
@end

Note: If the base class and subclass declare a "private" property or method with the same name it will compile without warning or error, but it will fail spectacularly at runtime as both classes unknowingly interfere with each other's private data.