There are currently three significant variations on 4 fin designs for wakesurfers. Inland Surfer offers a 4 fin on several of their boards, but the rail fins are the trailers. I've read where Jeff Page once referred to that as an A-Frame fin pod.
Smed, Shred Stixx, TWP, Stretch...sheesh the list seems endless all offer a more traditional quad with trailers that are inset from the rail fins.
Lastly, TWP is marketing something called "the stinger" which is a cross between a C-5 and a Twinzer. Leading canards and standard rail fins.
Custom shapers like Stretch, TWP, Props and the like can also create boards with a true Twinzer set up - leading canards that are an inch or so in front of the rail fins. Although, to my knowledge, no one currently offers that in production.
All 4 fins purport to offer more drive than a twin or thruster due to increased fin area and reduced drag. The Inland A-frame and TWP Stinger fin pods address the issue of "looseness" by placement of the extra set of fins. These designs have canards. In the case of the IS, they are inset towards the middle of the board and don't accentuate attachment to the rail fins, however, they can be fairly large and as such offer a great combination of drive, without the penalty of an extra long fin pod.
The TWP Stinger has very small canards and offers additional drive by increased attachment of water along the rail fins. The relatively small canards don't impact the "looseness" of the tail as much as a standard trailing fin quad.
Smed is doing some interesting work on quads and has shown a few boards utilizing fairly small trailers. Such a configuration offers the benefits of additional drive, with just a tad of an increase in the stiffness of the tail.
4 fins with a longer fin pod, WILL have a wider drawn out turning radius (think of it as the wheelbase on a car, a stretch limo has a larger turning radius than a mini cooper) than a comparable twin fin.
If I were to summarize, I would say:
Traditional quad with trailers and twinzers - offer most drive, and the widest turns. Traditional quad is most drivey and has the stiffest tail.
IS A-frame and TWP Stinger offer somewhat less drive than traditional quad and twinzer, but more than twins and don't sacrifice the loose tail and tighter turning. These are sometimes felt as "twitchy" to folks that prefer a longer
smoother turning radius.
The Smed quads with the small trailers are less drivey than traditional quads, perhaps a bit more drivey than the A-frame and Stinger, but it's close. Turning radius is on par with a standard quad, but the tail is less stiff and can be "broken loose" a little less than A-Frame, but more easily than traditional quad.
I'd concur with both Rob G and Derek, more slots for fins offers more potential options, although the Stinger would have the least available options as there are only two commercially available fins for the canards.