In a tiny house built for cold climates, how does steel framing compare to other materials?

The biggest concern with steel framing in cold climates is what is called Thermal Bridging. If you were to take a thermal camera to a stud wall (wood OR steel), you'd notice that the most energy is concentrated where the studs are. This means that there is a bridge of thermal energy transferring from the exterior of the house to the interior of the house or vice versa leading to possible issues such as poor cooling and heating abilities, condensation and, most frighteningly, mold. This problem is easily solved through the use of a Continuous Insulation (CI) layer. Typical wall assemblies that require CI would specify the use of something like 1/2" Polyiso sheet foam in 4' x 8' sections. This minor insulation helps stop the thermal bridging. Although you may have solved the thermal bridge, it is still necessary to insulate the wall cavity with a material of your choice. Most people prefer to use closed cell spray foam for the cavity insulation, though you can use anything from fiberglass batts to recycled jeans. It all depends on the R-value you are trying to acheive.