Nichrome wire has far, far higher resistance per unit length at a given cross section. It's an alloy chosen for that property.
The power dissipated in a resistor is the product of the resistance and current, so a larger resistance at the same current means more power dissipated as heat.
What's more important is somewhat of a subjective topic. The concept of duality is one way to frame it: every electrical thing has a dual. One of the more obvious examples: Capacitors are the dual of inductors. Everything that's true about inductors (ideal ones, at least) is also true of capacitors, if you exchange voltage and current, series and parallel.
There are also many electrical machines that have duals. For example, the common loudspeaker is driven by magnetic attraction or repulsion between an electromagnet (voice coil) and a permanent magnet. These are low impedance speakers (usually \$4\Omega\$ or \$8\Omega\$, meaning that an amplifier designed to drive them will be designed to output a large current over a small voltage. But, there are also electrostatic speakers which are high-impedance devices (\$>10M\Omega\$, easily), driven with small currents at high voltages. Rather than being a mostly inductive load, they are a mostly capacitive load.
The world is full of these duals. So as far as theory goes, mostly voltage and current can be exchanged and you end up with a different circuit or machine that accomplishes the same thing.
However, we live in a biased world. Voltage sources are more common than current sources. When we represent physical quantities electrically (like sound pressure) we tend to analyse them as voltages, not currents. When we think of a mechanical actuators, we think of magnetic solenoids before we think of electrostatic ones. I'm not entirely sure why this is true, but it is. Maybe it has to do with the practicalities of constructing things with the materials we know about. I've actually thought about framing it as a question on this site, but I couldn't think of a way to do it that wasn't too subjective.
Here's the lesson to take away: because voltage sources are so common, it's common to only need to consider the current, because the voltage has already been decided for you. If you have say, an Arduino that runs from a 5V supply, then you don't think about the voltage. The voltage is 5V, and there's nothing you can do about that, if you want to use that Arduino. All you can change in your design is how much current you require that 5V supply to provide.
However, there is no theoretical difference in importance between current and voltage. They are two sides of the electrical coin, equally important. And, in many cases, you can trade one for another. Consider both equally in your thinking.
Best Answer
You can keep increasing the voltage on a wire up to a point. Unfortunately higher voltage does require thicker wire, and it's nothing to do with heating.
The electric field at the surface of a single wire of given radius is proportional to the voltage/radius. Once the electric field exceeds the breakdown field of the dielectric that's insulating the wire, it begins to break down the insulation. This creates corona if the dielectric is a gas, like air of SF6, and damages the insulation if it's a solid.
This is handled on overhead wires by making a bundle of thin wires. This behaves, as far as corona is concerned, as a much thicker wire. Each wire is 'shielded' to some extent by the others, which reduces the electric field at its surface. While the 275kV grid tends to use a bundle of four wires, MV grids use several 10s of wires and a bundle more than 1m in diameter.
Many of us have seen pictures of high voltage switchgear. The conductors between nodes are often very large diameter tubes. This is simply to allow air insulation to be used with minimal corona