The actual standard voltage is 12 Volts. Some 12-Volt mikes can function with 9 Volt phantom powering, but then the maximum SPL is often reduced; it differs with different manufacturers and models.
External phantom powering always involves some voltage drop across the feed resistors, which doesn't apply to an internal 9-Volt battery (just so that we're not comparing apples with oranges here). The voltage of a phantom power supply is specified under no-load conditions, so by the time a microphone is connected and is drawing current through the feed resistors, the initial 12 Volts may become somewhat less than 9. That's what the microphone was designed to work with if it really is a P12 microphone--but if you start with only 9 Volts, then the voltage in the cable would be less than 6, which would be way out of spec. Due to that fact, and the way most DC converters work, the maximum output level of the microphone can be reduced by quite a bit more than the simple factor of 9:12 (~2.5 dB) or even 6:9 (~3.5 dB). Powering such microphones externally at only 9 Volts can really lead to premature overload.
But even if your particular microphones don't suffer too badly, I would never recommend a 9-Volt phantom power supply. Some day you may well have other and better microphones than you do now. Thus the remainder of this reply assumes that we're talking about standard 12 Volts versus standard 48, rather than (totally non-standard) 9 versus any kind of 48.
OK. For some microphones, 12-Volt phantom powering may be more efficient than 48-Volt powering, enough to make a real difference in the battery life of your preamp or recorder. For example, Schoeps CMC 6-- and CCM-series microphones can use either 12- or 48-Volt phantom powering, but their 12-Volt mode is nearly twice as efficient (48 V x 4.5 mA = 216 mW while 12 V x 10 mA = 120 mW); that's fairly typical DC/DC converter behavior. The same thing applies to Schoeps CMC 3-- (P12) vs. CMC 5-- (P48) amplifiers. Same audio performance but only half the power requirement? I'll take it.
This increased efficiency may apply to the recorder/preamp as well as the microphones in some cases, since 48 Volts is almost never required in a preamp or recorder for any other purpose (all in all it's a pretty odd choice of a voltage), and phantom powering in a preamp often uses more energy from the battery than the whole rest of the preamp circuit. I happen to be looking at the Sound Devices preamps these days (including their new MixPre-D) and am glad to see that they not only are switchable between 48 Volts and 12 Volts, but that when they're switched to 12 Volt powering they also use the right resistor value for that voltage. That's important, too!
--best regards