Here are the advantages/disadvantages as shown on UESP:
If Paarthurnax dies:
Arngeir retracts the Greybeard's hospitality, and it is no longer possible to ask him for the locations of Word Walls .
Paarthurnax will no longer be around to provide meditation bonuses to certain Shouts .
The Blades will welcome you once again, enabling the completion of Delphine 's recruitment quest.
Esbern will offer to take bones and scales in order to create a potion which grants a powerful permanent bonus to damage versus Dragons (currently, however, the potion is bugged and does nothing).
The Greybeards will not host a peace treaty, making the Civil War questline mandatory to complete in order to proceed with the Main Quest (if Season Unending has not been completed and the Civil War questline has not already been completed).
If Paarthurnax lives:
The quest will never disappear from the journal, and the dialogue options concerning the player's decision will remain for all the characters concerned. (This may be a glitch, as Paarthurnax is labeled essential after the Main Quest is completed). (Edit: Tell the Greybeards you wont kill Paarthurnax to remove from journal. Works sometimes, but not every time.)
Arngeir will still disclose the location of Word Walls.
Paarthurnax will still provide meditation bonuses for Shout words.
Paarthurnax will appear for an extra cutscene after Alduin's defeat, wherein he details his plans to rein in the Dragons and teach them to live under the Way of the Voice.
Delphine's recruitment quest will remain uncompleted.
You need to kill Paarthurnax to get back into blades. Some sources suggest that it is not possible to kill that dragon after the main quest, but regardless I have managed to kill it straight after the main quest line.
Wiki shows all the pros and cons for killing this dragon
It seems the results will not be very good at this moment, so just beware.
Here are results for kill Paathurnaax:
If he's dead,
The blades will greet you with warm welcome in Sky Haven Temple whenever you talk to them.
The greybeards won't be able to talk to you since you killed Paathurnaax. Arngier will say: "We have nothing to say to each other. Paathurnaax's muder is beyond my powers of forgiveness."
Because of the treachery to Paathurnaax, the greybeards won't host the peace treaty of the civil war and the dragon menace if you haven't did Season Unending and talked to the Blades instead of the Greybeards.
Since they can't talk to you, they won't be able to let you find out all the locations of the words of power because of the dialogue. Even if you force it with cheats, it still won't work.
Since Paathurnaax is dead, he won't let you meditate on word of power.
If you let him alive,
The blades will still tell you to kill him. The quest will be stuck in your journal. You can remove it by talking to the greybeards about what the Blades are up to about the dragon.
Arngier will still tell you about word of power now that you explore by yourself.
Paathurnaax will be at the end to the game talking about the prophecy of the dragosn and Dragonborn.
Esbern won't create the potion.
It's better to not kill him. The Greybeards care about the voice and and dragonborn and the prophecy. The Blades all they care about is the dragonborn and alduin and paathurnaax.
If you've already completed the main quest, you're actually unable to kill Paarthurnax. He's marked as Essential after you finish the main quest. If you'd like to remove the quest-log entry, you can try talking to Arngeir and telling him you don't intend to kill Paarthurnax - that should remove it from your journal, though reports are mixed about just how effective that is.
(Some users report that they are still able to kill Paarthurnax after completion of the main quest.)