The Nightmare' was the visual equivalent to the 'Welcome To My Nightmare' album. It was aired on US Television (ABC 11:30 EST 25th April 1975) to great acclaim. It takes the form of a long form 'video' to tell the story of the album, the first film of it's kind. All the songs from the album are featured plus 'The Ballad Of Dwight Fry'. The film also features horror superstar Vincent Price, performing his role as 'The Curator'.

The version of 'The Ballad Of Dwight Fry' is actually the original instrumental track with a new, probably live, vocal by Alice.

The 1975 'Welcome To My Nightmare' television special won an Emmy for best video tape editing in 1976.

Around 1984 'The Nightmare' TV special was released on Warner home video, just when the home video market was starting to open up. Due to this it became eligible for a Grammy that year, and was indeed nominated, but lost out to Duran Duran who were the current big thing. Alice attended the awards ceremony that year and presented with Grace Jones (Feb. 28th 1984). Original VHS copies quickly disappeared became extremely rare.
The film wasn't officially available again until August 2017, when it was finally re-released on DVD, coupled with a slightly edited version of the 'Welcome To My Nightmare' Wembley concert film ('Department Of Youth' is cut from the live film, although it's still listed on the back cover and on the DVD menu!).

The Show Producer, David Winter, went from arranging lavish TV productions for Rachel Welch and others to arranging even more lavish Alice Cooper stage shows.

.. was watching 'The Nightmare' from '75 last night, and wondering why Coop sings different words to 'Devil's Food', 'Some Folks', 'Only Women Bleed', etc. I'm talking totally different lyrics. Was it edited for T.V.?

Most of the changes were pretty small. Some of the lyrics ('Only Women Bleed') were definitely due to TV censors. It wasn't X-rated but it was still 1975 and long before people like Howard Stern came along and pushed the envelope to a point where you can actually say 'penis' over the air with only some people freaking out.
Some of the other lyrics were probably redone for the fun of it, or just too actually make it a separate vehicle for the video, and a little to tell the story of the Nightmare video differently (as in 'The Awakening').
(Renfield, October 1997)

From TV Guide April 25th 1975:

Wide World In Concert
In a departure from the usual format, 'Alice Cooper -- The Nightmare' features Alice experiencing a musical nightmare in which he is led by a spirit (played by Vincent Price) through a series of grisly encounters. Spiced with lavish dance numbers, the production is based loosely on his road show.  Among the numbers sung by Alice: 'Welcome to My Nightmare', 'Some Folks', 'Devil's Food', 'The Black Widow', 'Only Women Bleed', 'Years Ago', 'Department of Youth', 'Cold Ethyl', 'The Awakening', 'Steven' and 'Escape'. (90 min.)