As super means above and para means beyond, I would say the differences lie in the parts natural and normal.
In the mere word sense something paranormal is just beyond normal. Natural can be interpreted as synonym to normal in the sense of something comes natural and the like. It can be interpreted as another realm as well, the realm of God as opposed to the realm of man.
As @Aidas would use paranormal for the mysterious, one could argue that the realm of God is paranormal. But it is very unusual to talk of the realm and the workings of God as paranormal, while it is perfectly normal to call effects believed to be caused by UFO's, aliens (from outer space), or ghosts paranormal.
In addition it seems more fitting to call effects paranormal, while supernatural would also be used for beings.
paranormal is translated into German as übersinnlich, above the sense realm, while supernatural is translated überirdisch, übermenschlich, übernatürlich: above earthly, above humane, supernatural.
There is a common expression "to give (someone) the clothes off your back", meaning that you would go without basic necessities yourself in order to provide for that person; the Dursleys certainly haven't been doing that for Harry! JK Rowling is playing with our (the readers') presumed understanding of that expression.
His uncle is using "clothes" figuratively, as a stand-in for all the necessities of life; Harry's answer turns this around - his clothes are literally Dudley's hand-me-downs, but presumably the food he eats hasn't gone through Dudley first!
Side note: Clothes have a lot of significance in the Harry Potter books; house elves are automatically freed (and banished) if their owners give them clothes. Coincidence?
Best Answer
Put on is a phrasal verb, and transitive phrasal verbs such as this allow movement of the particle, on in this case, when the object is not a pronoun. If the sentences describe the act of dressing, then the meaning is the same.
There may, however, be a difference between British and American English in the choice between the two. The Corpus of Contemporary American English has 23 records for ‘put on my clothes’, but only 9 for ‘put my clothes on’. The British National Corpus has one record for each. These results are not necessarily conclusive, particularly given the virtually insignificant returns from the BNC, but they suggest a tendency which might merit further investigation.