What are the advantages of using the Wrangler?

The main advantage of the Wrangler (other than the enjoyment of controlling your sentry) is that it gives the sentry infinite range. An example: Imagine you have your sentry set up at the back of cp_gorge on the platform overlooking the final control point. From this position you can see the area between the two catwalks through the large metal door. Normally, this is far too far out, even for a level 3 sentry. However, using the wrangler extends the range sufficiently for you to attack the players coming through the door. It is used mostly in defensive situations when the sentry gun is situated out of range of enemy players who threaten the objective.

One other potential use (particularly useful in MvM) is the engineer version of rocket jumping, where the engineer uses the wrangler to shoot sentry rockets underneath his feet when he jumps, propelling himself to great heights or distances like a soldier or demoman explosive jumping. However, this is rarely used.

The Wranger has a lot of good going for it:

• You can choose your targets. Instead of having the sentry waste ammo into an über or a Bonk Scout, have it fire at someone else.
• You get a shield that blocks two-thirds of damage while aiming it. Have an über Heavy incoming? Turn on the Wrangler between wrench swings to keep its shield up while repairing it.
• You can fire at things beyond detection range. Are rockets, grenades, or Snipers harassing you from a distance? Use the Wrangler to return fire. You don't even have to worry too much about aim; the Wrangler has a subtle auto-aim function installed. You do have to be aware of damage-distance falloff though; wrangled sentries don't do full damage to things outside detection range.
• You can use your sentry to rocket jump (or bullet jump). Then perpetuate shenanigans, like building a base on the B or C roofs in Gravel Pit.

The Wrangler is at its best when you're only controlling it for one of these situations. Having it on manual all the time is not where it shines.

The biggest advantage is that a wrangled sentry has a shield which deflects about two thirds of damage dealt to it. This can be used to keep your sentry alive if it is under attack, for example if the enemy deploy an ubercharge.

The wrangler can also be used at any range or on any target- good against disguised spies or faraway enemies.

Finally, the missiles from a lvl 3 sentry can be used to rocket jump if you fire them at your feet. Accessing areas of the map that an engineer normally can't reach means you can put teleporters in unexpected places.