The restaurant you're referring to was actually called Tower of Pizza, although you're certainly not alone in making that error.
Located across from the Aladdin (now Planet Hollywood), near the Jockey Club and Lucky Ducky Liquor store, Tower of Pizza was a popular late-night hangout with mobsters, including Tony "the Ant" Spilotro, and Strip performers. It was originally owned by a well-known illegal bookie and loan shark from New York called Gaspare "Jasper" Speciale, a former client of Oscar Goodman whose name was finally added to the Black Book in 1989. You can read more about him in John L. Smith's On the Boulevard.
In 1979 Speciale sold the business to Bobby Capozzoli, the son of a California restaurateur who'd arrived in town earlier that decade. In a 2007 interview, he recalled how popular the joint was: "There were crowds three deep all the time," Capozzoli says. "Some of the entertainers who would come in at 3 in the morning would sing or play their horns." Oscar Goodman also has fond memories of the place: "We used to hang out there, with Bobby 'The Hunchback' Brent, Jasper, those were the good days," Goodman says. "All the characters used to come in there."
Then, when the rent almost tripled from $5,500 to $17,000 a few years later, Capozzoli moved the business to Boulder Highway and the Stardust became the new "mob" hangout (among other places).
In 1985, he and his brothers Joe and Mickey opened Capozzoli's at 3333 S. Maryland Parkway, which drew the crowd from the Tower of Pizza days once again and was the subject of the July 2004 LVA "Local Corner". We're not sure exactly when the latter closed -- Anthony Curtis recalls Tower of Pizza being here when he arrived in 1979, but he's not sure if it lasted to '85. Capozzoli's has since burned down -- twice -- the latter closure in 2007 turning out to be permanent.
Bobby Capozzoli had sold his share prior to this and bought an interest in Casa di Amore on East Tropicana (also a former Local Corner). Capozzoli later sold out to his partner and was considering retirement when he was approached by two cousins, one of whom was Liberace's former chef, who wanted to open their own joint and have Capozzoli host. This was DeStafano's, also on E. Trop, which closed in January this year. According to a message on usmenuguide.com, they hope to reopen at a new venue in the not-too-distant future.