There is no difference other than the price, at least on the plane. On board, the only difference between passenger service is differentiated by the class. Usually, there are up to 4 classes: Economy, Premium Economy (or Economy Comfort), Business, and First class.
I think there will also be no difference at the airport for Economy class at least. Some airlines reserve their lounge to business tickets for their customers only (so if you buy a business class seat with Air India for a Singapore Airlines operated flight, you might not be able to access Singapore Airlines lounge for example).
Other than that, ticketing conditions are usually different on a lot of points. First, you will not get the same amount of frequent flyer miles, or even none. It might also affect which frequent traveller program you credit your miles to (as @Gagravarr added). Refund and exchange conditions will be specific too. There might other ticketing conditions I forget that depend on the seller and not the operating airline.
Note: on Germanwings, I was asked my name to know the fare class I booked with, and was given a small bag with food whose content depended on my fare class. So in this case, passengers in the same cabin may have a different flight experience.