Unless of course it's not just the current that matters, and that voltage matters more than just producing a certain amount of current
In real circuits both matter. An IC, for instance, needs a certain minimum voltage to work (silicon diodes, for example, don't conduct at all below about 0.7V). On the other hand, too high a voltage may destroy semiconductor structures just by the force it exerts on the electrons.
Then, as per V=RxI, or I=V/R, higher voltage often forces more current through a given conducting structure, which technically is mostly not a problem by itself, but a higher current through any non-superconductor causes higher power loss in the conductor which heats the conductor and may ultimately cause irreversible thermal damage.
So, practically, we need to maintain the voltage between the lower upper bound for the circuit to operate. Too low and it won't work, too high and it may burn out.
As others have answered, most non-trivial circuitry does not act like a constant (ohmic) resistance but varies its apparent impedance over time. Digital ICs (CMOS), for example, often briefly consume much more power on the edges of their clock signal than in between, so powering via some sort of constant current source will force higher voltage and too much current through them at times and/or not enough at other times.