Imagine that you have the circuit below:
simulate this circuit – Schematic created using CircuitLab
Let's say that you wanted to catch the user pressing the button. The input signal will go from high to low, which is a falling edge. So you would use a FALLING interrupt to catch this event. Easy, right?
Now let's say you wanted to catch a user releasing the button instead. This time the inverse is true, so you want to use a RISING interrupt.
You should note that since these interrupts are edge triggered, the simple circuit below would not be ideal. Without any debouncing, there may be several edges to trigger from, resulting in repeated interrupts. Also, given the analogue nature of the design, we can't be sure that the rise / fall times will be fast enough to trigger the edge interrupt. We can get around this by adding a 100nF capacitor across the switch and following by a Schmitt trigger (which I can't find a symbol for in the circuit editor!).
So to answer your question, you use the interrupt that is correct for the event you are trying to catch. The Arduino MCU implements both edges for completeness and flexibility in your design, so if this interrupt was coming from an external IC you have no control over the design of, you can easily match the interrupt to its behaviour without having to add in an inverter.