Noise alone, has no meaning. In electronics circuits, you are measuring some quantities, voltage or current. So the SNR is the important value. If you have a noise of 1v it seams a lot, but compared to 100v signal - it insignificant.
About current and voltage noise, for a specific component as a resistor, those quantities are connected by ohm's law. So it is true that increasing resistor will increase Vn^2, but decrease in^2.
Where to use each of them: Vn^2 usually used in a cases with small impedances for example terminations in RF... also Vn^2 used as input noise representation of opamps...
Current noise is used with high impedance sources (current sources) like : high impedance detectors (photo/radiation detectors).
Basically it is the same and we use it in a way that it reduce our efford of calculation and better understanding of circuit...