What are the advantages of each? What situation would I want higher amperage than voltage, and vice versa?
You want a high voltage in e.g. electrical power transmission. The transmission losses are \$I^2 R\$. So, for the same power (e.g. 10kW) it is better to have 10kV and 1A than 100V and 100A.
For welding you want to have high current and low voltage, because the current determines the energy in the arc.
In digital electronics, where CMOS logic is used, you control everything just with voltage and ideally you don't want any current to flow. As you may know, to turn on or off the MOSFET transistor, you need to apply positive (or negative) voltage on its gate. In BJT transistors you need current because they are turned on with current applied.
To transmit energy on the long distances we use very high voltage. To understand why, consider the following
I is the current, U is voltage, P is power and t is time. We can calculate the heat Q according to the Joule–Lenz law. You can see, that to decrease heating you can either decrease the resistance R of the wire or increase the voltage. It is very expensive to decrease the resistance since you have to change the material of the wire or to increase its cross-section area.
However, you can just increase the voltage, because increasing voltage 10 times will decrease heating 100 times.