What is the difference between private and fileprivate in Swift 4

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>
Example: 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>
Example: private class SomePrivateClass {}

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

Answer to your question: (In Swift 3, private variables in a class are not visible in its extensions in the same file. For that, fileprivate had to be used.)

Yes, in Swift 4.0, Private is now accessible in extension but within same file. If you declare/define extension in other file, then your private variable will not be accessible to your extension

Look at this images:
File: ViewController.swift
Here extension and view controller both are in same file, hence private variable testPrivateAccessLevel is accessible in extension

File: TestFile.swift
Here extension and view controller both are in different files, hence private variable testPrivateAccessLevel is not accessible in extension.

Here class ViewController2 is a subclass of ViewController and both are in same file. Here private variable testPrivateAccessLevel is not accessible in Subclass but fileprivate is accessible in subclass.

Applicable in swift 4.0 and its versions
Private
Private access only in class and its extension(When extension is in the same .swift file).

File Private
File-private access only in class and its extension & subClass(When extension or subClass is in the same .swift file).