What are the advantage of the different gear shifters
This is based on my experience with the following type of shifters, but is limited to Shimano so others may have a different experience.  I haven't tried to split out the different variations of brifters either since I've only used 105's.

Integrated brake/shift levers (aka "brifters")

Pros:


Shifters are always close at hand, so you can always shift when you need to without taking your hands off the handlebar (unless your hands are on the flats).
This means shifting while climbing/descending is a breeze.
Similarly, brakes are always nearby


Cons:


Expensive
Complicated devices, so a mechanical issue can cripple your shifter when on a long ride far from the nearest bike shop.
Brake levers are very likely to take a pounding in a crash (whether major or minor), which is bad news for brifters (see "expensive" and "complicated" above)


Bar-end shifters (mount at the end of drop bars or aero bars)

Pros:


Simple devices, usually indexed for quick gear changes, but can also be switched to "friction" mode which allows you to dial in your gear by feel (handy when your shifting goes out of alignment).
Lower total cost than brifters (including separate brake levers)
Easier to learn than down-tube shifters


Cons:


Shifters are nearby, but you still have to take your hand off the bars.
Inconvenient if you need to shift while out of the saddle.
Possible to hit them with your knees (depends on bike and rider geometry) which can cause an unwanted shift (and can be quite painful)


Down-tube shifters

Pros:


Simple like bar-end shifters (including choices of indexed/friction shifting)
Also inexpensive
Shorter shifter cable runs
Not likely to hit your knees


Cons:


Quite far from the bars, so you'll likely shift less often than with the other styles
Inconvenient for shifting while out of the saddle or during a white-knuckle descent
Takes some getting used to finding the levers without looking
Not all modern frames will have the appropriate braze-ons for mounting (but most should).


My current bike uses bar-end shifters based on some of the reasons above.  Plus for long randonneur rides (over 20 hours) I lose a lot of hand strength and the bar-end shifters are the easiest to work with (can shift using the heel of my palm).