What are the sonic differences between…

Paradigm Sub 1 (2012) and a Velodyne DD 18 (2007)?

Maybe I'm wrong, and since I don't have any subwoofers, I may definitely be.

I thought that you weren't supposed to hear a subwoofer, that the low frequencies that it's putting out is under the hearing threshold and basically just the air and pressure of the long waves.
Krell_man, you may be thinking of the oft-repeated admonition that you shouldn't hear the sub as a separate entity. It should extend the bass of your speakers transparently, without calling attention to itself as a separate driver.

Or, perhaps, to the fact that few instruments have musical fundamentals below about 40 Hz, although some do, e.g., pipe organs go all the way down to 16 Hz.

In any case, you can hear the lowest frequencies a dynamic sub puts out, at least down to 14 Hz (the shibboleth about being able to hear only down to 20 Hz apparently isn't true). Also, depending on the installation, subs may be used through the entire deep bass range, e.g., crossing over at 80 Hz or even in the midbass. This is typically true when they're used to supplement small satellite-style speakers that are specifically designed for use with a sub.

Finally, subs produce a lot of out-of-bandwidth signal because the crossovers aren't completely sharp. Forex, if they cross over at 80 Hz, you'll still be hearing them at 160 Hz with a four-pole crossover or even higher.

Bottom line, they're very audible.
"I thought that you weren't supposed to hear a subwoofer"

It's not that you don't hear the subwoofer, you should not be able to localize it in the lower registers. Believe me I sure do hear the punch and impact of my Revel B15 in the 30 to 60Hz range.

As far as the Sub 1 and DD 18, I am sure both are top tier in punch, impact, and overall musicality.

"Stealth Sub" Cannot be heard. ;)
Stealth Sub, cannot be heard, or seen on radar. I'd opt for the Velo. I have 3.
I kept my questions general because I wanted a full range of responses. (But, regardless of the forum involved, I get tired of the responses which begin with: “It depends…” or “I have not tried it, but… ”.) I already have an Infinity Intermezzo 1.2s. I have an opportunity to purchase a used Velodyne DD 18 (2007) for $1,700 or purchase a new Paradigm Sub 1 for about $2,200. This conundrum is a matter of opportunity.
My point was, if the subwoofer didn't make an audible sound and just put out the waves, how would you know which one was better? It would make a difference in that the less distorted waves would be the better of the two, but I didn't know if that could be easily determined.
The answer is that the waves are quite audible. The differences in subs are audible as well and can range from subtle (when comparing very good ones) to immense (when comparing for a example a good one to a boomy home theater one).
Okay Up-since you refuse to provide the application and you don't like "it depends", here's my next best answer. Ideally, all subs should work the same, with the only difference being how much they load any given room. Having said that, I would choose the Velo 18 because it would never strain for impact. Connections can be another difference, as some don't provide speaker level options. I own four different subs, two for HT LFE (separate systems), one for HT center channel extension (speaker level) and one in 2-channel vinyl-based system. I can hear and feel each sub in its application but cannot identify sub location.
One more thing, I used to own a Velodyne and really liked the quick, tight, deep punch in HT application.
I understand why you wanted to keep this general, but can you at least say whether the sub will be used in a HT, music only, or some mix thereof? Might help surface some useful information on one or both of the subs you're looking at, or maybe introduce some other potential options. Just a thought, and best of luck in your search.
Here is my equipment:

Pre-Pro: Anthem AVM 50V
Amplifier: Anthem PVA 7
Center: Paradigm CC570 v.3
Fronts: Paradigm Studio 40 v.2
Surrounds: Paradigm Studio ADP-590 v.1
Rears: Paradigm Studio 20 v.4
Subwoofer 1: Infinity Intermezzo 1.2s
Subwoofer 2: Paradigm PS-1000
Subwoofer 3 (new, unopened, not in use): Def. Tech. Supercube 4000
Source Blu-ray player: Oppo BDP 95
Projector: Panasonic PT-AE4000U
Power management: PS Audio Power Plant Premiere AC Regenerator

I have a vaulted living/dining room that is 24’ long and 15’ wide. The screen is at the front (8’ end) of the room and the listening position is about 12’ away, with the remaining 12’ behind the seated area. The vault is 16’ high, at the back of the room, behind the seating area. The floors are hardwood and the walls are sheetrock. We have two sofas, two bookcases, a table, a buffet and some things on the walls. The main use of the HT system is for movies. The sound, especially the dialogue, is not as clear as I would like (i.e., not as clear as the theater I built in the basement of my old house). I have an opportunity to get a subwoofer at a great price: A used, 2007 Velodyne DD 18 (with room correction) for $1,700, or a new Paradigm Sub 1 for about $2,200. In reality, most of the problems with the sound are due to the acoustics of the room, but who can pass up a deal like this for one (or both) of these subwoofers?

More... The front, right half of the room opens on one side with a ½-high wall to a closed hallway, and the back, right half of the room opens to another hallway and also to a second open area to a den. Basically, the back half of the room - with the vault - is open.

The CC is directly in front of the seating position and is tilted up about 30 degrees to aim directly at the listeners’ ear. The CC, fronts and Infinity sub are on a similar line facing the seating area, with nothing in between them and us. There is a large rug on the floor between all the speakers (including the CC) and the seating area.

The suggestion of buying one Sub 1 now and getting a second one later is a good idea. I like having two subs and having them matched will probably improve what I am hearing/feeling.

The last issue, that I didn’t want to muddy the waters with, is that we will probably be moving in 6 to 12 months. Unfortunately, we don’t know where we are going. But at least, we will be taking our HT equipment with us (again :)