9v Power vs Wall Wart Power

So my project is too complex to explain here, but basically I am experiencing different results using the 9v and the wall wart on my uno. My project involves a relay, but the coil is not connected in any way to the arduino. When using the wall wart, the relay experiences erratic behavior such as clicking on and off rapidly at times and not registering some inputs. I have found others on this forum with the similar problems using the wall power, but they have just been pestered with possible errors in their circuits. Therefore, I would simply like to ask: What is the difference between using a 9v(yes, I know it’s terribly inefficient) and the wall wart to power the arduino?
I would specifically love to know if at all the 3.3v or 5v outputs, ground ports, or analog inputs are effected in any way.

Thanks!

The answer would depend on what type of 9V battery, the current and voltage characteristics of the wall wart, all the other circuitry and finally, the code.

Generally 9V battery isn't enough for anything involving a coil, beit motor/servo/relay, since
they are designed for a few tens of mA current drain.

But there's no point generalising further, I've got no data to go on.

I assume that's a 9V battery?

What's the wall-wart's voltage? And make sure it has DC output, not AC.

My project involves a relay, but the coil is not connected in any way to the arduino.

What is connected to the Arduino?

Does the regular Blink LED sketch run normally from the wall wart with nothing else connected to the Arduino?

I would specifically love to know if at all the 3.3v or 5v outputs, ground ports, or analog inputs are effected in any way.

If the power supply is also 9VDC there should be no difference.* However if higher voltage is applied, there is more voltage drop across the 5V regulator and it will dissipate more heat. So for example, depending on how much current you are pulling from the Arduino's 5V pin the regulator may overheat with 12V applied, but be OK with 9V applied.

* I'm assuming the 9V battery is "holding up" (the voltage is not dropping under load). And, I'm assuming the 9V wall wart is regulated at 9VDC (not just 9V "nominal".)