A good deal of debate exists with respect to what the stage building in Greek and Roman theaters looked like. In both the Greek and Roman theater, the skene referred to a structure (a stage building, if you will) at the rear of the "stage". In Latin, the term frons scaenae means the "front of the scaena", so, technically speaking, this would be different from the Greek skene, which denotes not only the front of the structure, but the whole of the structure.
Both the Greek and the Roman structures were probably similar in that they had, at most, three openings that could serve as "doors" to various "houses". Both structures also allowed for performance on more than one level. So, both the Greek skene and Roman frons scaenae had some flat, horizontal space above them that would allow an actor to stand several feet above ground level. Modern reconstructions of Roman theaters, however, make the frons scaenae a two-storied structure with a flat roof, whereas the Greek skene, at least in the fifth century B.C., appears to have been one storey with a flat roof.
Of course, the Greek orchestra was round and was the primary place where the actors moved and danced, whereas the Roman orchestra was semi-circular and was a place where dignitaries would sit and watch the plays (Vitruvius 5.6.2). Given this, Vitruvius tells us that the Roman frons scaenae was further back from the orchestra than the Greek skene.
Finally, the Greek word skene means "tent", which implies that the Greek skene was not a permanent structure. In fact, I am not aware of any Greek skenai that still exist. There are, of course, numerous Greek theaters that have been excavated, but, as far as I know, the skenai that are preserved there were built later by the Romans.
In the early days of Roman theater, the skenai were also not permanent, but, beginning in the first century B.C.E., the Romans did begin building permanent theaters, so the theaters that we think of today as Roman do have permanent and quite elaborate scaenae.