What if V+ = V- with an OpAmp?
simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. OP's schematic redrawn with more realistic power supplies.


  In the two simulations below, Vout is almost the same (to GND) in both while the OpAmp sees the same differences in potential at its inputs, so actually, the second output should be 1V lower. How is that possible?


You have two big problems with your schematics:


The supply voltages are way too low - particularly for a TL081. You will see this in the datasheet.
You have no feedback on your circuit so the op-amp output will be the "open-loop gain" (> 1,000,000) times the difference between the two inputs. You have shorted them together and you are probably expecting a zero output as a result. Real op-amps aren't perfect, however, so the very slight differences between the two inputs - known as the input offset voltage - will cause the output to swing as hard as it can towards one supply rail or the other depending on the polarity of the difference.


With such low supply voltages the output will probably get stuck at about half-supply voltage.

Note that simulation results will vary with the quality of the simulator. Some will give realistic simulation of details like input offset voltage and various other non-ideal characteristics. Operation at very slow supply voltages may be one of the conditions that may not be simulated well.