Referring to the schematics in the question, there are unlikely to be any advantages to using an op-amp, and quite possibly a number of disadvantages.
First, your op-amp schematic is drawn without feedback and with the negative input grounded. Due to the high DC gain of the op-amp, the output will most probably be saturated at either the positive or negative rail (even if you can adjust the potentiometer to just the right position, the resistor's temperature coefficient will probably make the circuit too unstable to be usable).
Second, although the op-amp can output a voltage close to the supply rails, the potentiometer circuit allows for slightly higher voltage.
That said, there are ways you could use an op-amp as a buffer to drive the motor, which would reduce the fluctuations in the motor voltage due to load on the motor. This is because the back-emf produced by the motor will fall as the motor is loaded, and thus the current drawn will increase. Without the op-amp, this changes the effective resistance of one leg of your potentiometer voltage divider.
TL081 is not a particularly good choice for low DC voltages (it's better for AC coupled designs). Precision op-amps have offset voltages in the 10's of microvolts or lower, and there are even auto-zero op-amps that have negligible offset voltage, an drifts in the tens of nV/°C (at some significant cost in other characteristics). It's also quite noisy (25nV/\$ \sqrt{Hz}\$), but at least has a typical flicker noise corner frequency in the 1Hz range. Aside from high input Z, it has basically one really nice advantage, it's really cheap, and widely available, which is why I've actually designed such a 37-year-old op-amp into a new product recently.
A good general purpose (low noise, capable of handling +/-15V supplies, low distorion for AC signals) precision op-amp suitable for mV levels might be the OPA209A.
You can certainly null the offset voltage of your TL081 out using a trimpot as shown on the datasheet, but it won't stay that well nulled for long. A 10°C change will typically change the offset voltage by 100uV, and about one out of every two will be worse (there's no guarantee how much worse, but a guess would be most are better than +/-30uV/°C). An OPA209 is going to be roughly an order of magnitude better.
There are probably going to be better (and many worse) choices for any given application, all things considered. It's amazing what performance you can get for a a few dollars, so it's worth looking around rather than trying to make a silk purse of a sow's ear.
Just to give you an idea of the kind of (in) accuracy you could get, consider that the gain of the TL081 is only guaranteed to be >15,000, so a gain of 1000 amplifier could have a gain error in the 6% range even without the input offset error (which would be very temperature dependent, and has a -3dB corner of something like 20Hz. Cascading two \$\sqrt {1000}\$ gain amplifiers would help with that (null only the first one).
If the range is, say +/-5mV input, frequency is 0.001 to 1Hz and required accuracy 1% of FS + 50uV**, it might be typically *** okay in a lab environment with a light output load, if you null it after warm-up.
** Instrumentation type specification- it means the output could be as much as +/- 100mV from the ideal value with any input, so a 1mV input could give you 900mV or 1100mV.
*** "Typically" means that one chip might be okay, and the next might not meet the requirements. Guaranteed value is probably 10 times worse.
Best Answer
Your formula is written such that it assumes that Vdd and Vss are symmetric with respect to ground. If this is not true — e.g., you are using only a single power supply with Vss tied to ground — this formula does not apply.
If you want to use this circuit with a single supply (or asymmetric supplies), you'll need to connect the grounded end of R1 to a "virtual ground" at (Vdd+Vss)/2. In fact, what you can do is simply split R1 into two separate resistors with twice the value, and connect one between R2 and Vss and the other between R2 and Vdd.