What are the advantages of the angled flight deck layout used by some aircraft carriers?
This kind of layout is called the angled flight deck (also called 'skewed' or 'canted' deck) and was first used by Royal Navy. There are a number of advantages to this layout compared to using the same direction of landing and takeoffs:


It allows concurrent landings and takeoffs.
By having an angled deck, the aircraft which fail to connect (with the arrestor cables) during landing can go around and try again without risk of damage to other (parked and taking off) aircraft).


The main reason for the emergence of angled deck was the introduction of jet aircraft in naval operations, which due to their weight and high speeds required most of the deck available to complete an arrested landing. Also, in case of missed wires, it was possible to use a barrier in case of propeller aircraft, which was not possible in case of jet aircraft. According to Rear Admiral Dennis Campbell, the inventor of angled deck,:


  But with the advent of the heavier faster jets, it soon became obvious that the dimensions of the carrier's deck would impose a limit to the standard of aircraft that could be operated. The inevitable increase in weight and landing speed of the jets would make it impracticable to retain the existing combination of arrester wires/barriers/and deck park without a drastic reallocation of the deck space required to absorb the much longer pull-out of the wires. 


Prior to the application of the angled deck, this caused significant problems and requirement of complex machinery to overcome.


  Ark sister ship and predecessor, the Eagle, was equipped with sixteen arrester wires and three different types of barrier (together with all their several hydraulic machines), and all this gear could only just cope with the first generation for (straight wing) jets.  No way would such a layout be good enough for the newer types.


The solution was to skew the flight deck by about 10 degrees (the HMS Ark Royal first had the flight deck angled by 5 degrees, which was then increased to 10) and use it for landing. This lead to:


  
  The saving in arrester gear units and barriers – Ark needed only four wires and one (emergency only) barrier.  The reduction in weight and the extra space that this conferred enabled more mess-decks to be fitted in, thus reducing congestion in living spaces.
  Since landing accidents would obviously be far less frequent, fewer aircraft and spare parts would be needed so costs would be greatly reduced. 
  Similarly, death and injuries to aircrew and deck handlers could be kept to a minimum – with a big boost to confidence and morale, not to mention the saving in training costs.
  As experience was gained of the new scheme, less obvious advantages came to light.  New areas of parking, (abaft the island for instance); an added flexibility in the relative wind speed and direction, easier and faster deck handling, less wear and tear on the command, etc.
  


Perhaps surprisingly, though the USN has adopted this system in its supercarriers, the RN dropped it later on.