Ms Giuffre claims she was trafficked by Jeffrey Epstein and forced to have sex with Andrew when she was a teenager, which the duke categorically denies.
In an interview with BBC Panorama, she told how Andrew was "the most hideous dancer I've ever seen in my life" and "his sweat was ... raining basically everywhere".
She added: "I implore the people in the UK to stand up beside me, to help me fight this fight, to not accept this as being OK.
"This is not some sordid sex story. This is a story of being trafficked, This is a story of abuse and this is a story of your guys' royalty."
The hour-long investigative programme entitled The Prince and the Epstein Scandal is being broadcast on BBC One on Monday at 9pm.
In clips released ahead of the screening, Ms Guiffre alleged how she was brought to London by Epstein in 2001, introduced to Andrew and taken to Tramp nightclub, and asked by the duke to dance.
She told Panorama reporter Darragh Macintyre: "He is the most hideous dancer I've ever seen in my life.
In his recent Newsnight interview, Andrew claimed he had a medical condition in 2001, after suffering an overdose of adrenaline in the Falklands War when he was shot at, which meant he did not sweat.
When they had left the club, Ms Giuffre said the socialite Ghislaine Maxwell, Epstein's then-girlfriend, gave her instructions.
"In the car Ghislaine tells me that I have to do for Andrew what I do for Jeffrey and that just made me sick."
She said that later that evening she had sex with Andrew upstairs at Maxwell's house in Belgravia.
The duke has denied meeting Ms Giuffre, then called Virginia Roberts, saying: "I can absolutely categorically tell you it never happened. I have no recollection of ever meeting this lady, none whatsoever."
The duke has been pictured with his arm around Ms Giuffre's waist at Ghislaine Maxwell's house.
He said he had no memory of the photo being taken and questioned whether it was his own hand in the image.
Ms Giuffre told Panorama: "The people on the inside are going to keep coming up with these ridiculous excuses.
"He knows what happened. I know what happened and there's only one of us telling the truth, and I know that's me."
Andrew's attempts to defend himself against Ms Giuffre's previous accusations and explain his friendship with convicted sex offender Epstein in a BBC Newsnight interview last month were branded a "car crash" and led to him withdrawing from public duties.
He was widely criticised for failing to show remorse for his association with the disgraced financier or empathy with Epstein's victims during his television appearance.
He has resigned from a large number of patronages and seen organisations he was once involved with sever ties.
Ms Giuffre claimed in court papers in Florida that she was forced to have sex with Andrew when she was aged 17, below the state's age of consent.
Buckingham Palace has branded the allegations "false and without any foundation", stating: "Any suggestion of impropriety with underage minors" by the duke was "categorically untrue".
Ms Giuffre alleges the duke slept with her on three separate occasions, twice while she was underage.
During his Newnight interview, the duke said an alleged encounter with Ms Giuffre in 2001 did not happen as he spent the day with his daughter, Princess Beatrice, taking her to Pizza Express in Woking for a party.
Epstein took his own life in August in a New York prison while he was being held on sex trafficking charges.
Ms Giuffre's interview took place before the duke's appearance on Newsnight.