The advantage of a regulated power supply is stability and power line immunity. You can use 2 stage regulation. It will be very quiet without a lot of capacitors. In addition the voltage to the amplifier will be pure, low impedance DC no matter what frequency the amplifier is drawing keeping frequency response of the gain stages independent of power supply reactive elements, i.e. capacitors. Also, keeping the heater voltage stable keeps the operation of the tube stable. Sonically it should be punchier in the bass.
RC smoothing can be very quiet at high frequencies (hiss) but power line variations can show up in the output of the mic, as I wrote in an earlier post. That is noise, just not hiss, but subsonic garbage nonetheless. If you're lucky it will be very low level. In addition, the current draw of the amplifier, being AC, will interact with the passive power supply. A cap's impedance goes up as the frequency goes down thus the power supply becomes part of the amplifier stage.
Sonically I prefer to work with everything well regulated. In multistage amplification Isolating the B+ for each stage provides stable imaging keeping the instruments where you put them in the mix.
The buzz may be your power supply has poor grounding. Go to a negative regulated filament supply. A LM7906 is a good choice followed with a 1000mfd cap, bypassed with a 1 mfd mylar cap. If that's not quiet enough you can do double regulation.
Both B+ and Filament share the same ground wire in the cable, which means one can modulate the other if there are varying ground currents. The way around that is either send independent grounds to the mic or tie the grounds together at the output of each power supply. Varying current causes varying voltage due to the resistance of the wire.
HOWEVER!!:
That said, I need to see a schematic of your mic with parts values and voltages and see how the tube is used, please. I need to know how many milliamps of B+ the mic will draw.