I think rather than forward- or down-firing, you're more likely to hear a difference between sealed versus ported subs, with some people who will tell you that sealed subs are more "musical". Then again others will tell you that a well-made ported design can sound very clean also (which I would agree with). But the sealed subs tend to require more power to hit higher sound levels, and therefore tend to be more expensive. The SVS and Axiom subs are both ported.
I haven't heard the Axiom subs, but based on the Axiom speakers I do own I'm sure their subs perform well. In the end, I went with the SVS because it was the most affordable way to get clean bass all the way down to 20hz. The SVS subs are great for music, none of that mid-bass bloat that some cheaper subs use to give the false impression of powerful bass. Where the SVS really excels is in giving that last octave of performance, which really adds extra oomph for movies with lots of LFE.
The advantage that the SVS design has is enclosure volume. By using the cylindrical shape, the SVS has the capacity to hit low frequencies at high SPL's without lots of distortion and without requiring thousands of watts to get there. The cylinder also has the advantage of being very strong/rigid, so they don't have to use a bunch of bracing like you do to build a rigid cube subwoofer.
I'll admit the aesthetics are a mixed bag; some people like the look, and some people don't. The thing I like about it is although it's very tall, my 20-39PCi doens't take up a lot of floorspace. A cube with similar performance would take up considerably more floor-space (unless you're going to spend a fortune on one of those high-performance small subs such as the Sunfire that Alan mentioned).
