With the release of Java 8, there is now a third option.
Runnable is a functional interface, which means that instances of it can be created with lambda expressions or method references.
Your example can be replaced with:
new Thread(() -> { /* Code here */ }).start()
or if you want to use an ExecutorService and a method reference:
executor.execute(runner::run)
These are not only much shorter than your examples, but also come with many of the advantages stated in other answers of using Runnable over Thread, such as single responsibility and using composition because you're not specializing the thread's behaviour. This way also avoids creating an extra class if all you need is a Runnable as you do in your examples.