What is the benefit of a curved up flight deck?
Because the method of launching aircraft are completely different in these carriers. Carriers with ski-jump, initially designed for RN Harriers (as @Simon already has explained), requires that the aircraft take-off under its own power. 

On the other hand, large aircraft carriers having flat decks have their aircraft launched by catapults (with a notable exception) which basically throw the aircraft from the deck. Though this system is heavy, expensive and complicated, it brings significant advantages to the table:


It allows aircraft with higher t/o weight to operate from the carriers. USN operates F/A-18E/F Super Hornets, which weigh ~30 tons from its carriers (they even operated the heavier F-14 Tomcats). The ski jumps, on the other hand restrict the aircraft t/o weight (which means less ordnance), as the Chinese are finding out. Indian Navy operates Mig-29Ks from its carrier, but they are lighter (~25t).
The catapult launch offers more options - it can operate in wider range of sea conditions compared to the ski-jump and can even launch aircraft even if one of the engines fail during the launch, a feat usually not possible for the aircraft in ski-jump.
The catapult allows for a significantly higher takeoff rate, which means a higher sortie generation. As rusnavy.com notes:



  With its four catapults, a US carrier is capable to shoot one aircraft in every 15 seconds. Kuznetsov has only three takeoff tracks. Moreover, two bow tracks are not intended for fully-loaded aircrafts! They can take off Kuznetsov only from one track which starts far behind the midship, i.e. an aircraft must run almost the whole flight deck! Comparing to catapult, takeoff rate of ski-ramp is at least twice lower. 



The catapult allows the launching of support aircraft (like AWACS), which have low thrust-weight ratio and would not be able to take off under their own power. 
The ski jump takes up space in two ways- physically, it occupies front deck space (on the other hand, aircraft can be parked over catapults when not in use), while the take off run required by the aircraft (greater with more gross weight) eats up more space, leading to limitations of the number of aircraft on the deck.


As for the exception, the USN operates carriers without ski-jumps, but these are not called as such- they are called amphibious warfare ships an operate only VTOL aircraft like the F-35 Lightening II and V-22 Osprey.