This contest makes it quite explicit that you answer EXACTLY as it says to answer a question. It said "First it was a supper club. Then, a burlesque club. These days, it's a country club-occasionally. What is the name on this famous 50-something's neon, set in a town know for it's tunes?".
The correct answer IS The Continental Club. But the question expressly asked "What is the name on this famous 50-something's neon". So, you must look at the actual neon sign. It is spelled with all capital letters for both THE and CLUB. Continental has just the capital "C".
So, again, it is like this-- "THE Continental CLUB".