most of it is in the name. the internal wastegate is actually inside of the exhaust housing for the turbo and the opening of it is controlled by a waste gate actuator. the exhuast gas that flows through the internal gate is automatically dumped into the exhaust. the external wastegate is mounted somewhere on the exhaust manifold and is strictly controlled by boost controller(or spring inside the gate) and a boost refference via a barb, usually on the top. the exhaust that flows through the external gate is usually vented to the atmosphere through what is know commonly as a dump tube, but can also be routed to the exhaust to cut back on the noise although it usually causes a loss in power routing it to the exhaust.
both are used to vent exhaust away from the turbo to be able to maintain a set psi designated by the user.
interal gate are usually common to lower power applications and are relatively good at controlling boost when you aren't pushing the turbo to its limits. boost is controlled via a wga and the amount of boost that is able to be held is determined mostly by the size of the turbo, but also the quality of the wga.
external gates are usually common to bigger setups but are versitile because you can use them on any application. external gates are usually more expensive but they allow you to control boost a lot better and even with smaller turbos, you will be able to hold more boost towards redline than you would with an internal gate setup. the tricky part, but not really, is getting the right size external gate for your setup. too small and you will encounter what is known as boost creep. too big and you won't be able to control boost very well be cause it will constantly be fighting to stay shut.
that is all i can think of right now. hopefully some more people will chime in.
Moved on to bigger and better things. the ride: SRT4 with 230 whp stock!!! i loved the focus but this one is just a better car for what i want. not so stock anymore. pushing 307whp/301wtq @ 19 psi.
Midwest Crew #14