What is the difference between private method and private extension

Are you talking about 'private' and 'file private'...?

With Swift 4.0, scope of

private

and

fileprivate

is extended.
Private is now accessible within same file, outside actual class/extension. If you declare/define your extension in other file, then your private variable will not be accessible at that time fileprivate will work

File Private
File-private access restricts the use of an entity to its own defining source file. Use file-private access to hide the implementation details of a specific piece of functionality when those details are used within an entire file.
Syntax:

fileprivate <var type> <variable name>

fileprivate class SomeFilePrivateClass {}

Private
Private access restricts the use of an entity to the enclosing declaration, and to extensions of that declaration that are in the same file. Use private access to hide the implementation details of a specific piece of functionality when those details are used only within a single declaration.
Syntax:

private <var type> <variable name>

private class SomePrivateClass {}

Here is more detail about all access levels: Swift - Access Levels

There's no difference between marking an extension private, or marking individual declarations within an extension private. It's a matter of personal preference. (However, you can't declare an extension private if it's adding a protocol conformance.)

The details are here:

https://docs.swift.org/swift-book/LanguageGuide/AccessControl.html

The section on extensions is near the end.

Finally: member "privacy" is applied across all extensions in the same source file. A private member declared in one extension is usable by all members of the type declared in the file, not just those declared in the same extension. This wasn't true until Swift 4. Before that, you had to use (the old equivalent of) "fileprivate" to get this behavior.