Even with the mistaken 43F capacitors in the applet, I don't see how it can produce that output with the opamps working across purposes like that. However, if I flip one of the opamps over so it has positive feedback, then the circuit amplifies the input by about 3x (albeit while oscillating at 2MHz.) I suggest double-checking your reverse engineering. (I wouldn't trust that java applet to be the truth.)
Ohh thanks for pointing out the capacitor values, changing them gave me a voltage swing of about ~700 mV with the Java applet which seems to correspond with the VU meter swing. I checked my reverse engineering and both inverting inputs are used for the feedback and both non-inverting inputs are used for the inputs.
Here are pictures of the top of the PCB and another with a superimposed trace pattern. There's a pair of LEDs in parallel at the output which I haven't bothered to put in my simulations since they only appear to be there to clamp the output voltage (I'm still trying to get the output to oscillate in LTSpice). I get the same results with the LEDs in the simulation. I've also omitted the balancing potentiometer as the simulation shouldn't need it due to the ideal nature of the components.
PCB Component Side
PCB with Superimposed TraceTwo pins of the leftmost header are used for traces that went to a light bulb, the next three are for ground and part of a +/-V supply circuit. The right header is used for the differential (?) signal input. Here's a pinout of the dual op-amp IC:
LT1057/LF353N PinoutI don't know where V(n007) is, but note that the output resistor is also floating, so you'd want to measure across it or give it a ground reference.
It's the top of the output resistance with respect to ground. Looking at the voltage only across the resistor the waveform is identical but doubled since there's the contribution of the lower op-amp. Also, I was mistaken before, the Java applet is indeed providing the voltage across the output resistor.
Not sure if this is what you are seeing, but on LTSpice components have an orientation, so check your resistors to make sure current is not going backwards:
The orientation is only a reference thing. For example, if you're expecting a current to go one way but have the simulation assuming the other, it will show a negative current. It has no impact on the analysis.
Edit: Here's the trace side of the PCB; the superimposed picture had some confusing artifacts from the post-processing.
PCB Trace Side