Ok, this has been hashed out more than any other topic in Miata circles. Visit miataforum.com and do a search in the Power Mods forum for more than a fair dose of discussion.
Here's my take:
For most top-end power (ie, most hp), you'd go with a large turbo. Larger turbos take longer to spool and don't have the greatest low-end response, but make huge power later.
For low-end grunt, a supercharger or a small (particular ball-bearing) turbo will yield better results. Small BB Turbos can be into boost under 2k rpms and most roots-type (screw) superchargers start about there as well.
Mrclam, I'd venture to say that a roots-type supercharger has more moving parts than a turbo. In any case, superchargers generally deliver a more linear power increase, since they are "always-on". But they can also suffer from decreased thermal efficiency as well as parasitic drive loss (from the belt). Superchargers are difficult to intercool well, resulting in maximum boost of around 8-10 psi.
In modern turbo systems, there really isn't the lag that most people refer too. My miata, with its small bb turbo is on boost instantaneously, in fact, it's hard to keep it out of boost.
Both types of systems, with over anything past 6 psi, will require an integrated fuel delivery and timing management system. This could include, either more fuel pressure mapped to boost/vacuum or more fuel volume with larger injectors and electronics. For timing, you need a retard controller and/or knock sensor. Both of these could respond to any combination of boost, rpm and knock sense.
There's no really easy answer except that for HUGE power, you need a turbo. For most of us, the decision really rests not on Turbo vs. Supercharger, but against what kits are available for the car you're looking to FI. You need to weigh the advantages and disadvantages of those kits against each other.
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Tony
'01 Spectra Blue
'94 Turbo Miata