In some of the designs, power output is limited by the power supply. In the US, powering a circuit designed for a low-current/high impedance 9V battery with a high current/low impedance power supply could violate FCC regulations.
You don't want a power supply or battery to have an internal resistance.
The power supply probably made a hum or buzz noise in the transmitter because it wasn't filtered enough.
A battery will give less signal noise relative to most power supplies, noise would come from the power supply as Audioguru stated.
My post was merely informational, using an antenna different than design, different power supply, and other considerations of a transmitter can put the effective radiated power to the point it interferes with neighbors. The .gov won't come knock your door down (usually), unless it is a commercial product or gross negligence. At the same time, people don't like interference if they are trying to get a DX station (Far away/weak signal).
Something else, most oscillators of this type are extremely sensitive to power supply variations, so an extremely slight ripple becomes exagerated beyond what it would normally be. A battery will drift over time, but slowly.
to help eliminate the hum, try using putting a 220 uF cap across the pos and neg of the power source, it acts like a filter. Also, use an RF choke in series with your power input. That should take care of it.
Power supplies aren't perfect. Here is what a simple power supply does:
They take the AC from an outlet which then leads into a transformer. This transformer steps the 110V down to a lower voltage (say 12V for example; it depends on the number of turns on the transformer) then it gets rectified from diodes. These diodes chop off the negative portion of the 12V AC signal. Diodes have what we call "insertion loss," so some of the voltage is reduced because this loss. So usually a transformer will do 110V to 12.6V because the diodes have a .6V loss. Then you have a capacitor. When the 12V is at the peak the capacitor is charged, then when it is not at the peak the capacitor discharges but makes it a lot more steady and straight. So simply put, the capacitor reduces ripple.
Now if you put that power supply that I just described on a oscilloscope, you would still be able to see a small amount of ripple whereas a battery doesn't have this ripple.
Bugs are illegal, but Frequency Modulated Wireless Microphones are not. Go figure. Also, on poorly designed transmitters, the frequency will tend to drift as the battery dies, unless you use lithium batteries.
Battery produces clean dc voltage while power supply after being rectified may still produce some ripples which will cause humming at the output. Also if PWM is used inappropriately, radio frequency noice may occur which may affect other electronic devices connected to the same power supply.