There can be benefits to both in different situations. And in reality, as you said, we never have a pure 'voltage' or 'current' - you have both. The key is the impedance: A high output-impedance source will have it's output current change very little if the load changes, but the output voltage will change a lot - you want your load (say an amplifier) to measure the current. A low output-impedance source will have a very small change in output voltage when the load changes, but the output current will change a lot - you want your load to measure the voltage.
Take for example the inside of an integrated circuit. If we have a voltage signal, what we actually mean is 'the voltage between two points' - usually this is between the signal wire and the ground. However, the ground is not necessarily stable - other currents are also flowing through this ground and might cause voltage drops, which will lead to errors.
A current signal will not suffer from this moving ground - a current of say 1 mA is still a current of 1 mA if you have a bit of difference in ground. (this is incidentally the same reason why industrial sensors like using current mode sensors - the long wires can cause a significant voltage drop that leads to errors if you were to use voltage-mode signalling).
There is also the matter of impedance. Some sensors have a high output impedance, such as a photo-diode. Because of this, these sensors are usually used as a current-source, and not a voltage source.
On the other hand, at low speeds, a MOSFET appears like a high-impedance input voltage-to-current (the 'correct' name is transconductance) amplifier. You want to drive it with a voltage, as it will turn this voltage into a current at its output. So when driving a MOSFET, you want a voltage signal.
Things change when you go to systems with matched characteristic impedances at higher frequencies. At that point, every device has the same impedance, and you don't really think about voltage and current anymore, and instead consider power.