Why did Betty ask about the smell when they got to the hotel in Rome?  Diesel or Gas or Rubber, I think she said.

Hmn, I didn't think much of it besides maybe a conscious attempt from the writers to knock some reality into Betty's escapism? A lot of people who haven't been to postcard perfect places haven't yet smelled them and as a consequence do not realize that it might be more polluted, crowded, stifling, and perhaps even smellier than whitebread subruban North America.

At least that was my take.

It might be a simple dose of reality that they were in another land. 

We went to Rome on our honeymoon 14 years ago there was a distinct gas/pollution type smell where ever there was lots of traffic.  Europe didn’t have the smog requirements we have in the states and you could tell.  Although all emission requirements were the same in 1963 (none) motor smells should have been equal, however Italy and Europe generally had crapper cars that results in all the smells that come with engines burning oil/transmission fluid etc.

My family owned two older European sports cars in my youth (Fiat & MG) both were quite smelly

Thanks for the insight, having never traveled overseas, I didn't know.