It has been suggested that the many answers be combined into a single CW answer.
This is that answer.
Feel free to update with any additional information.
Advantages of Gateways: Rally point can be set.
Queuing of units is allowed.
This means you have more time to remember to train units, reducing downtime for some novice players.
It also means production of the next unit can begin immediately after the last one has finished.
Units can be canceled.
Units can be created using only hotkeys (no mouse clicks).
Units cannot be attacked until they are spawned and ready to attack (whereas units warping in can be attacked during the warp in period).
Advantages of Warpgates:
Units can be spawned quickly anywhere you have a pylon.
Queuing of units is not necessary -- you do not commit resources until moments before the unit is spawned.
Army composition can be decided at the last minute.
Cooldown period is incurred after the unit has been produced, which can be an advantage during a battle.
Cooldown period is shorter for warp gates, allowing you to train more units from warp gates than from gateways in the same amount of time.
gaming.stackexchange.com/users/816/sjohnston
Gateways can be rally-pointed.
With warpgates, you must click the exit location every time you create units.
gaming.stackexchange.com/users/816/sjohnston
You can queue them up and forget about them.
Handy if you're a beginner and have lots of resources.
gaming.stackexchange.com/users/2578/BlueRaja - Danny Pflughoeft
You used to be able to turn them back into gateways in between warp gate cooldowns to produce ~twice as many units.
This is no longer possible, as the conversion time back and forth was increased.
The only remaining advantage they have is that you can queue multiple units up at a gateway.
gaming.stackexchange.com/users/336/Raven Dreamer
One that hasn't been mentioned is that you don't need pylons to create units at a gateway.
So if you're heading into enemy territory, want constant reinforcement, and don't want to mess with creating a pylon or flying around a warp prism, you can just waypoint your gateways to one of your units and have all your new units follow along as they're created.
Granted, that's a lot of ifs.
gaming.stackexchange.com/users/2831/Kyralessa
Besides the other answers, one minor advantage a gateway has over a warpgate is the enemy can't kill units being constructed by a gateway.
gaming.stackexchange.com/users/1912/ILMTitan
One advantage I can think of is to reinforce small or medium groups of harassing troops on the go.
Being that you can set the rally point once for all exiting units and queue up production, you could very well set up a small army of say 5 to 10 units with constant reinforcements to harass an isolated expansion with little or no defense.
I don't know though how I would fit this into a mid game scenario, since warp prism or proxy pylons would work far better in reinforcing 'on the go'.
gaming.stackexchange.com/users/3071/wormintrude
Advantages of Gateways: Ability to queue units If they are bound to a hotkey, units can be created without mouse clicks Advantages of Warpgates: Ability to create units anywhere a pylon exists.
If someone backdoors you with a nydus worm, you can warp in units to deal with the threat.
If you have a forward pylon, you can instantly reinforce the front line.
gaming.stackexchange.com/users/908/Equistatic
I believe there is a 10 second cooldown advantage so you would gain 8 seconds there alone.
Now add the time it takes to walk across the map......
huge advantage on warp - micromanagement overhead though...!
gaming.stackexchange.com/users/None/
Gateways let you CANCEL units.
I can't say how many times I accidentally warped in a ton of sentries instead of stalkers because of a bad keypress, and I couldn't cancel it.
Warpgates incur cooldown period after the unit has arrived, not before.
gaming.stackexchange.com/users/6774/tenfour
I don't think anyone mentioned it, but the cooldown on warp gates is a few seconds less then gateway build time for each unit.
This means, if you're optimal on your W usage, you can pump more units with a warpgate then you could for a normal gateway.
gaming.stackexchange.com/users/6296/eighteyes
Once in a game I used the warp prism to deploy a smal force on the platform where my opponent had his starting base (without destroying the garbage layed out before the main entrance).
Before he realized that it is not-so-unreachable as he thought my five warp-gates released a nice army to overwhelm his base.
But for mass-production the normal gateways are more usable.
gaming.stackexchange.com/users/3506/OlafW
In StarCraft2, Protoss can change their gateways into warp gates. However, I'm puzzled that they can be changed  back  into gateways.  Is there any advantage for gateways over warp gates ? What am I missing ?
Are there advantages for Protoss gateways over warp gates?
gaming.stackexchange.com
<starcraft-2><starcraft-protoss>
gaming.stackexchange.com/questions/7485
gaming.stackexchange.com/users/3436/Leonel
Gateways can be rally-pointed and can be queued up and forgotten about. There is also a 10 second cooldown advantage. The main advantage is that you can queue units created in them, which are invulnerable at that time. You can also use them to reinforce other units or make rally points. Another advantage is the fact that they have a cool down advantage. 
Rally points can be created with them The ability to queue units is mentioned. The cooldown is also smaller. Reinforcement and invulnerability are also important advantages.