That depends a little on what one means by “costs.” In pure dollars and cents, it may look like it is cheaper to tear down an old building and build a new one, but that depends somewhat on what is replacing the old building, the level of rehabilitation the building needs, and its condition before rehabilitation.
The costs of tearing down a specific building varies significantly from one to another. It depends on the size, construction materials (think wood versus poured concrete) and if any environmental abatement is needed (i.e. asbestos). Those are just a few of the cost considerations when a building is torn down.
Similarly, new buildings have a wide variety of costs, too. Again, this depends on the size of building, use, quality of materials, type of building, etc.
Chances are that the new construction would not be built to the same quality as the building it is replacing – many of the historic materials cannot be sourced today (like tight-grain, old-growth lumber, real stone, etc.) and the craftsmanship of those buildings is mostly a lost art.
Think of a typical “pole shed” replacing a historic two-story brick commercial block – it’s not an apples-to-apples comparison. (More like marshmallows to apples.) When you take into account the environmental impacts, and how long the payback period is for new “green” buildings, renovating old buildings is almost always FAR more cost effective than demolition and building new. The National Trust for Historic Preservation GreenLab has done a lot of research on this topic. You can read their recent reports.
This explanation from a gentleman who commented on a MinnPost article, sums it up extremely well: “Virtually any commercial building can be retrofitted to be more energy-efficient for less money, and equally important, with less consumption of new resources, than razing the old building and starting from scratch. Paying big bucks to tear something down, wasting the resources consumed to build it in the first place, only to replace it with something only marginally more efficient than a retrofit of the existing structure, requires developers (and sometimes, taxpayers) to pay TWICE for the same structure, and it makes no sense on either fiscal or environmental grounds.”