I agree with you on Springsteen and my wife was startled by his appearance on some show, I think Jimmy Fallon. IIRC his voice was much higher than she expected.
Did you mean “Big Me”? I guess since the video was so big and visually associated with them I’ve never really thought about it that way. Though I guess it’s a much different style of song than “This is Call” or “I’ll Stick Around.”
Cab Calloway. He really drew out his “R’s” while singing, not so much when speaking (although his Blues Brothers segments may not be very representative of either).
Rather obviously, Yma Sumac.
Also, Johnny Winter.
Geoff Tate from Queensryche. very deep speaking voice but he sounds totally different when he sings. Met him at a toy show years ago, he collects vintage Matchbox vehicles.
I listened to an old radio program with Freddy Mercury being interviewed, from not long before Live Aid. He didn’t talk anything like his singing voice. If I hadn’t have known I wouldn’t know who it was being interviewed.
He actually talked like Rami Malek. 
I guess that’s good acting. Give that man an Oscar.
I was surprised to hear Emmy Rossum (of Shameless) is a trained opera singer. Her speaking voice is not what you typically hear from a singer, it’s rather low-pitched and nasally. But she’s got pipes! Check it out: Emmy Rossum Sings Opera For A Hot Dog | CONAN on TBS - YouTube
Country singer Dwight Yoakam. Back when I was a Country DJ 25 years ago Dwight was in his most hot period, and we played his songs all the time. We ran a feature in which Country stars were interviewed and when I heard his I was floored. He had absolutely nothing of that Country twang or accent in his voice,which was soooo prominent on his recordings.
Someone mentioned Bob Dylan and I was thinking the same thing when I saw the topic. I listened to Dylan’s music for probably 30 years and never once actually heard him speak. When finally I did I was surprised his speaking voice is nothing at all like his “singing” voice.