Ok, so bear with me - I've had this conversation before, I have a good idea of how it goes, so I'm going to ask your patience with the full answer ok - in the long run, this will be shorter for you, I promise.
Only one of the phones in this discussion have a microUSB port.
The rest all have an MHL port.
When MHL was introduced in 2011, the manufacturers all said that the phones that had them, had MHL ports.
And despite that fact, the popular press and people on forums who argued with that fact (it's not MHL, I can prove it, it works on my PC!) led to sales confusion. If you can't beat 'em, join 'em, so the manufacturers took to lying and saying that the phones had microUSB ports and by the way you can do stuff to get HDMI.
MHL used a microUSB connector on one side and a smart switch on the inside.
It physically detects what it's plugged in to and switches between USB circuitry and specialized digital audio/video (AV) circuitry.
That's MHL.
Unfortunately, the naming downgrade has led people into thinking that AV signals are going out courtesy of USB and the little MHL to HDMI rigs are operating on USB signals.
And that has led to a land office business in bogus, useless cables that physically cannot do what they claim to do.
Now - without a catalog reference, I can't tell if your cable is real or bogus.
If it's real, the point of the large USB cable is to supply power (using a standard phone charger) to the phone's MHL circuitry and especially to the micro-electronics embedded in the shielding and connnectors and those micro-electronics are converting purely digital HDMI into composite video and analog audio. (Per my first link.)
If it's bogus, you got bupkus.
I hope with all sincerity that your brother didn't get ripped off - but personally, I wouldn't apply power to that thing without knowing more to begin with.
If it ok - your S3 Mini will output nothing. It doesn't have an MHL port, it has a microUSB port. No AV output there.
Your S4 is an 3rd-generation Samsung device - 11 pins for Samsung accessories, not the MHL 5 pin standard.
The cable, if it is not bogus, is almost certainly a 5-pin type. Monoprice.com sells 5 to 11-pin converters, no clue if it will work with that cable.
Those are $4 and out of stock, have been for a while.
http://www.monoprice.com/Product?c_id=108&cp_id=10833&cs_id=1083314&p_id=9716&seq=1&format=2
For another 7 bucks - $11 - you can just get a powered MHL/HDMI adapter cable, with a separate 5 to 11-pin adapter included, and have the advantage of full resolution HD from your Samsung S4 instead of component SD video with that other cable (if not bogus).
http://www.monoprice.com/Product?c_id=108&cp_id=10833&cs_id=1083314&p_id=10021&seq=1&format=2
The cable you have looks similar to this -
http://www.amazon.com/Micro-Composite-Adapter-Galaxy-Android/dp/B00MD72NTY
If it's truly that one, it may be for real (with zero customer reviews, no way to tell) - but it came with a 5 to 11-pin adapter (not typical for the rip off imitations) and your S4 is going nowhere without that.
But it absolutely needs power to work.
Hope this helps, sorry if this sounded too pedantic but I didn't know another way to get this all straight for you without a lot of back and forth and confusion ok.
