Do the words paranormal and supernatural mean pretty much the same thing? ?

The words paranormal and supernatural and fiction mean the same thing. 

Paranormal is more specific in that it invokes science, ie; things that are beyond the scope of scientific understanding. It’s just a made-up term to try to legitimise pseudoscience. Things that are claimed to exist when there is no scientific evidence for them. 

"Paranormal" is an awkward term. I like the word Arthur C. Clarke used better - "strange." A strange thing is something that may or may not exist and, if it does exist, may or may not be easily explained. The word is used for all sorts of things that arouse skepticism - ghosts, UFOs, etc. Isaac Asimov, a skeptic, described supernatural entities as entities that are not subject to the Law of Conservation of Energy. In other words someone who gets along fine without ever eating. 

Skeptik hits the nail right on the head. The only thing I would add is belief in both paranormal and supernatural is faith-based. Both are believed in without proof or even convincing evidence.

Things that are called paranormal are things that "we" don't understand but may understand in future as we continue to study them. For instance, we now understand thunder and lightning has nothing to do with Gods having chariot races in the sky and throwing lightning bolts at us. Supernatural is something that is completely outside of our learning abilities as it is completely outside of our own understanding and always will be.

EDIT, Judging by my paedo detector, I see Sol is with us. Oops!, it looks like I upset her? HAHAHA!

EDIT. As you see, Gary is first to point out to people what "paranormal" means when it suits him. Or as you see here, it is now a made up word??? again, when it suits him. He will even deny this is the paranormal site if it helps his pointless argument. Nothing to be learned from that little waster. You should also check up on his other silly comments. Corkers like " there was no paranormal then, as the word paranormal did not exist then"???? Based on that logic, a newborn baby does not exist if it has not been given a name yet??? How about "People who believe in paranormal are closed-minded"? Wow!

Possum, Time and time again, we are told, when there is a ghostly presence in the room, all the heat (energy) is absorbed by the entity, not all the ham sandwiches. Also, a meter of what we call empty space contains enough energy to boil all the oceans of the world dry. So no need for a ham sandwich for the entity to exist. 

Though there is quite a bit of overlap in how the terms are applied, they don't technically mean the same thing.

Supernatural refers to things that inherently don't follow the laws of nature.  Things like deities, ghosts, demons, magic, miracles, etc.

Paranormal refers to things that, if they were true, would actually follow the laws of nature.  Just potentially natural laws that haven't been discovered yet.  Things like cryptozoology, aliens, some kinds of ESP, reincarnation, NDEs, etc.