Technically the cost of the actual rail infrastructure is less. It requires much less width for the corridor and both road and rail need roughly equal strengthening for the foundations. Overhead lines and signalling are not a significant incremental cost so long as you have an existing network.

However, you generally only build rail where there is sufficient latent demand. This means the land you require is of higher value and land is one of the most significant costs assuming you would need equal infrastructure requirements (e.g. bridges, structures, tunnels, etc.) regardless of mode.

Therefore, on a per kilometre basis, rail is often more expensive. The key difference is throughput. Rail is highly efficient for both freight and passenger movements, over sufficient distances, because it is a fixed corridor with right of way or full segregation. A dual track railway can carry far more tonnes of freight or many multiples more people than a dual lane road as it doesn’t suffer from congestion.

The other comments correctly mention aspects like managing terrain and the width of railroads vs roadways. What I want to highlight is the development of road building methods at around the same time that metal-on-metal rail developed.

The 1800s were a wild time. Some clever folks figured out that they could put a contemporary steam engine – invented early 1700s; used only for stationary uses in lieu of water power – onto a wagonway. Wagonways are basically wooden or metal guides/flanges so that a horse-drawn wagon could be pulled along and stay perfectly centered on the path.

Up until this point in history, the construction of graded, flattened surfaces for moving goods didn’t change very much compared to what the Romans were doing with their roads. That is, a road had to be dug down and some soil removed, then backfilled with coarse material (usually large stones), and then a layer of smaller stones to try to approximate a smooth surface. The innovations the Roman introduced included a keen eye for drainage – freeze/thaw cycles destroy roads – and surveying methods (also to build things like aqueducts and canals). And concrete, of course.

But even the best built roads of that era were still prone to rutting, where each passing wagon slowly wears a groove into the road. Wooden wagons wider or narrower than the groove would suffer poor performance or outright break down. The wagonways sought to solve that issue by: 1) forcing all wagons to fit within the fixed guides on the sides, and 2) concentrate the grooves to exactly within the guides. The modern steel-on-steel railway takes this idea to its logical end.

An adhesive railroad seeks to be: all-weather, heavy duty, and efficient. Like Roman roads before it, all railways (except maybe on-street tramways) need to excavate the soil and build it up, usually being higher and wider than the rest of the land. It also minimizes the width of the earthworks, by being so compact and building upward. This sturdy base also provides a strong foundation to support heavy loads, preventing the steel rails from sinking or “rutting”. And finally, putting the wheel atop the rail makes for low-friction operation. Early wooden plateways sort-of did this, but they didn’t manage curves like how modern rails do.

All the while, instead of trying to support heavy wagons, another clever person sought to reinvent road building outright, postulating that if a surface could just spread out the load from light/medium traffic, then the soil beneath could be used as-is, saving a lot of earthworks. A gravel surface would meet this criteria, but gravel is not all-weather and can develop rutting. The key innovation was the use of binder (basically glue) to hold the surface together, such as tar. This sealing process meant the surface wouldn’t shift underneath traffic. This neatly avoided the issue of dust, made the surface water impermeable, and reduced road maintenance. So famous is this surfacing process that the inventor’s name can still be found in the surface for airport runways, despite runways always being excavated down to a significant depth.

So on one hand, rail technology developed to avoid all the pitfalls of 1700s roads. On the other hand, road surfacing developed to allow light/medium traffic roads to be economically paved for all-weather conditions. Both developments led to increased speed and efficiency in their domain, and networks of both would be built out.

Rail networks made it possible to develop the “streetcar suburbs” around major historical cities in the late 1800s. But on the same token, cheap road surfacing made it possible to build 1950s American suburbs, with wide, pedestrian-hostile streets sprawling in serpentine patterns. The fact that sealed roads are water impermeable has also substantially contributed to water pollution, due to increased rain runoff rather than absorbing into the underlying soil.

If you need the bare minimum connection, a road is cheaper than a railroad. At scale, a railroad is cheaper than a road.

I am struggling to find a source, but the tradeoff between road and railways scales with congestion. The infrastructure needed for moving 1000 trucks per hour on a road is much more expensive than 1 truck per hour. Rails, however, scale much more easily, as the freightline is typically closely managed already.

Not the question asked, but relevant: When each individual enterprise considers its own transport needs, road transport is usually cheaper. However, when looking at the collective needs of an entire economy, rail is usually a way more efficient and cost-effective option.

Private rail companies will only invest where there are epic amounts of cargo or passengers to move, which when left to the private sector leads to massive under investment and over-reliance on road transport. There is no coherent argument against having extensive government investment in rail.

This is not something that is even a little debated. Rail is more efficient. Ship is even more efficient but since most of those are coming from redic far if your comparing overseas to local that is a different issue. But mile for mile and ton for ton should be ship, then rail, then road.

1. Road use is subsidized, as the trucks who weigh much more do pay a little more in licensing and gasoline to fund road maintenance, but nowhere near to the level of their added wear-and-tear.

2. Rail use has no subsidy.

There’s the math.