BJT, Resistor, and Diode Work but ULN2803 Does Not – Troubleshooting Guide

The piezo buzzer use piezoelectric effect to emit sound (the material in the buzzer deform itself due to voltage, deformation will be "proportional" to the voltage applied). Since sound is pressure wave one have to provide oscillating voltage to the buzzer in order to produce oscillating deformations. And most important:

From the electrical perspective piezo buzzer act like capacitor!

So why circuit with BJT and without R2 did not work?

With BJT in the configuration that you use, BJT can only "sink" current. That is it can only lower voltage at its collector. So after first "turn on" the voltage at the collector dropped to ground (or close) level. Next turn on cycle will not change voltage across buzzer as the voltage at the collector was already at ground level. (That is why you have measured 5V across the buzzer). Since voltage across buzzer do not change buzzer will not vibrate and thus will not produce sound.

Adding R2 solved the issue. When BJT is conducting, it lowers voltage at collector (C). When BJT is NOT conducting, R2 pull voltage up. This crates oscillation and produce sound.

Now, more interesting question is why MOSFET seemed to work (and BJT did not)?

I am not sure but here is my thory: My guess is that you have used one of this big power mosfets. They tend to have big parasitic gate to source/drain capacitance, at the same order of magnitude as a capacitance of the piezo buzzer(hundreds of pF). This parasitic capacitance created AC coupling between your input signal and buzzer which was enough to produce sound.

Now BJT of the type you have used have much smaller capacitance (single pF) that was not enough to provide enough AC coupling (it did provide some that is why you did hear some barely audable sound)

Note: The whole issue is little more complicated then simple AC coupling, but I believe is good enough first order explanation of the phenomena.

As Matt says, the first thing that jumps out is more capacitance needed.

Breadboards are terrible for stray inductance/capacitance so you really need to try and keep wires as short as possible, keep sensitive signals away from higher current/voltage/fast changing signals, and add plenty of bypass and bulk capacitance around the circuit.

I would start with a couple of >100uF electrolytics on each power rail and near the FET, with a few 100nF or higher ceramic caps on the power pins of the ICs.

Another thing to look at is using a separate rail for the relay, and routing the ground return separately from the uC's ground. Of course they need to be tied together, but you can ensure the high current return does not flow through the uC board's ground this way.