Not all mainframes are ancient; new models are still designed and sold to this day. And the brand spanking new mainframes may still be running COBOL code and other such antiquities, as many new mainframes are installed as upgrades for older mainframes and inherit a lot of legacy software that way.
And to answer your question: a mainframe is just a server. A specific design-type of server with a particular specialism for a particular set of usecases, but the basics of the underlying technology are no different from any other server. Old machines (mainframes or otherwise) will always consume far more power per instruction than a newer machine, so any old mainframes still chugging along out there are likely to be consuming a lot of power comparable to the work they're doing.
The value of mainframes is that they tend to have enormous redundancy and very high performance characteristics, particularly in terms of data access and storage. They're the machine of choice for things like financial transactions, where every transaction must be processed almost instantly, data loss is unacceptable, downtime nonexistent, and spikes in load are extremely unpredictable. For a usecase like that, the over-engineering of a mainframe is exactly what you need, and well worth the money over the alternative of a bodged together cluster of standard rack servers.
See also machines like the HP Nonstop line of fault-tolerant servers, which aren't usually called mainframes but which share a kinship with them in terms of being enormously over-engineered and very expensive servers which serve a particular niche.