In short the answer is YES.
The Samsung Galaxy S3 HDTV adapter works with the Samsung Galaxy S3 & S4.
The S4 HDTV adapter works with the S3 as well.
The S3 HDTV adapter has a 24Hz refresh rate whereas the S4 adapter has a 60Hz refresh rate. This is the key difference between the two. The S4 adapter creates a sharper on screen image.
The s4 does not need power to be connected for either MHL adapter to work, but it is advisable to avoid excessive battery drain.
When using either the S3 or S4 adapter with the Galaxy S3 you do need to have mains power connected.
See both adapters in action below.