Electronic – Op-Amp Sample and Hold Circuit Help

This should work, but usually there's also a resistor from the non-inverting input to ground:

If \$\dfrac{R4}{R2} = \dfrac{R3}{R1}\$ then:

\$V_{OUT} = \dfrac{R_{3}}{R_{1}} \times (V_{IN+} - V_{IN-})\$

To minimize offset error both inputs have to see the same impedance, and therefore \$R1 = R2\$ and \$R3 = R4\$.
Omitting R4 will only give a scaling factor for the voltage on the non-inverting input, namely:

\$V_{OUT} = 2 \times V_{IN+} - V_{IN-}\$

but changing the setting for the lower potmeter should definitely have a result in the output. Did you measure the voltages on both inputs?

What happens if you set the lower voltage to 2.5V and the upper to 1V? The inverting input should also be 2.5V, and the output 4V. What do you measure?

Note: especially the lower voltage follower is not necessary in your version; the opamp's input current is low enough to be negligible for most uses, and by the way, you're connecting the potmeter's wiper to an exactly same input!

Further reading
differential amplifier tutorial (interesting site overall!)

A peak detector is something of a sample and hold that samples all the time, and holds the peak:

Follow the input with this, and connect the output to your ADC. Make C significantly larger than the capacitance used by your ADC's sample and hold, or follow it with a buffer, so the voltage across C doesn't sag as you read it.

Trigger an ADC reading just after the event happens, and when it's done, close the switch to reset the peak, or make the switch a resistor forming an RC time constant significantly longer than the time it takes to measure the peak but significantly shorter than the interval between peaks to accept a little error but avoid the need to reset the peak detector.