Escape is better known as “The Pina Colada Song”.
Ob La Di Ob La Da is often called “Life Goes On”.
*Mambo #5 * is probably better known as "A Little bit of . . . "
Actually Escape is The Pina Colada Song. Official title = *Escape (The Pina Colada Song). * So does this actually fit into this thread? I believe the OP was looking for songs with misperceived titles, not songs with dual titles called by one or the other. That’s how I took it anyway.
(The Pina Colada Song) is like Episode IV: A New Hope. It was added retroactively, and isn’t part of the real title. The title, as it appears on the original album (which my dad has and I have verified personally), is simply Escape. The addendum was added later to aid those who were unaware of the song’s true title in finding it.
Except it’s Careless Whisper (no “s”) and it’s not by Wham!. 
OK, I am now informed that it was released under the Wham! name in the US, for some reason. In the UK it was a George Michael solo effort.
Becase, at least in the US, it was on Wham’s album Make it Big.
There’s quite a lot of these. Among them:
“Silent Running” by Mike & the Mechanics (“Can you hear me?”)
“Run Like Hell” by Pink Floyd
“The Logical Song” by Supertramp
“The Load Out” by Jackson Browne (technically part of a medley called “The Load Out/Stay”, but they can be perceived as different songs on the album Running on Empty)
“The Wedding Song” by Paul Stookey
“Ode to Billie Joe” by Bobbie Gentry
“Tubthumping” by Chumbawumba
Loads more here.
I’ve always called that song Never Gonna Dance Again.
I believe that’s “Won’t Get Fooled Again”.
How about The Drunken Dickhead Song as Jimmy Buffett’s Margaritaville is also known? Or is that just the former karoke host in me coming out?
Pink Floyd’s “Another Brick in the Wall (Part 2)” aka “We Don’t Need No Education”
In my house, Bill Withers’ “Ain’t No Sunshine” is known as “The ‘I Know’ Song” due to the long section where he sings, “I know I know I know I know I know I know I know I know I know I know I know I know I know I know…”
We called Take it Easy the “Runnin’ Down the Road Song.”
I learned that last year, when I moved within spitting distance of said bridge.
Most people of my generation know Scott Joplin’s The Entertainer as The Sting
How about Nazareth’s “Hair of the Dog”? I always think of it as, “Now You’re Messin’ With a Son of a Bitch.”
And to bring this thread to a new level, most people incorrectly refer to The Queensboro Bridge as The 59th Street Bridge. Note that there is no 59th Street in NYC – it’s called 59 Street…the th is just how it is pronounced.
How come nobody ever calls the Williamsburgh Bridge the Delancy St Bridge?
I should also add Synchronicity II by the Police. My first impression of it was “A Million Miles Away.”
Back when the Greaseman was a force to be reckoned with (at least in the DC radio listening area), the “Can Can” was popularly known as “The Tuna Fish Song.”
There is a Frankie Valli and the Four Seasons’ song that’s officially named Opus No. Somenumber-or-other, and before I found that out, I was never certain what exactly to call it. “Don’t you worry `bout me”? “I’ll be strong/I’ll try to carry on (although you know it won’t be easy when you’re gone)”? Usually, I would just sing the chorus, which features both of these phrases.