Sheffield City Council said the cladding at Hanover House was not made of the solid aluminium panels ordered.
The issue was spotted during fire safety tests carried out in the wake of the Grenfell Tower blaze.
Councillor Jayne Dunn said: "We will not stop until we get to the bottom of what happened."
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The council announced on Monday that a single element of the cladding system, added to the 16-storey tower block during a recent refurbishment, had failed new fire tests.
Hanover House resident and tenants association member John Cawthorne said: "I personally feel betrayed."
Ms Dunn, cabinet member for neighbourhoods and community safety, said the issue had been discovered following independent testing of the cladding at the council's 24 tower blocks.
She said testing had been carried out despite paperwork stating "there was absolutely no cladding apart from aluminium on our buildings".
"It's only when you cut in to it that you can see the layers," she continued.
"I can guarantee you that we will not stop until we get to the bottom of it because they took a piece of cladding to the tenants at a consultation and they didn't put that up.
"That was the cladding that we asked for. That was the cladding that we were told was on the building."
Mr Cawthorne said: "The Tenants Association were given samples, which I kept for eight years, and those samples are 3mm aluminium sheet.
"That's what the council asked to be put on this building, but that's not what was discovered on this building when the tests were carried out.
"People are angry and I personally feel betrayed."
Tests are being carried out on high rises across England in the wake of the Grenfell Tower fire in London on 14 June, in which at least 79 people died.
So far 75 UK blocks have failed fire safety tests in 26 areas.
An inquiry has begun after cladding removed from a tower block in Sheffield was found to be different to what council bosses had asked to be fitted.