The key difference is that the black leads on the simple Voltage Probe (VP-BTA) are connected to ground of the interface when used. That means that if you use two of these voltage probes at the same time, the two black leads are connected to a common electrical terminal and hence to each other.
In addition, the input impedance of the Voltage Probe is much lower. This means that the Voltage Probe will work well with battery-and-bulb kinds of experiments, but not circuits with large resistors such as RC circuits.
Here is the classic situation where this difference matters:
Imagine that you want to use two voltage probes to measure the voltage drop in a simple series circuit with two resistors. You put one voltage probe across each resistor. If you use the Differential Voltage Probe or Go Direct Voltage Probe, everything will work fine. If you use the Voltage Probe, you could have a confusing situation. If you connect the Voltage Probes as most people would, with the red leads each at the more positive ends of the respective resistors, then you will get very confusing results. The reason is that the two black leads of the simple voltage probes are both connected to the ground of interface, and hence to one another. You are shorting out one of the resistors. One voltage will be zero and the other will be changed because of the one resistor being shorted out.
It is possible to handle the situation above to get good results using the simple voltage probes. The way to do that is to connect the two black voltage leads to the same point on the circuit.
The bottom line here is: If you are going to be doing a lot with electrical circuits and you want to keep things simple, consider getting the Differential Voltage Probe. If you’re doing battery-and-bulb experiments only, you can probably get by with the simple Voltage Probe.