A ginger bug is simply a lactic acid culture started from raw ginger root (with skin still on) and sugar mixed together in dechlorinated water. When you "add the ginger bug" to your drink recipe, you're adding the liquid from this culture after straining out the chopped ginger bits. After the ginger bug has been allowed to mature to a slightly fizzy state (usually after three days), the strained liquid will be sufficiently saturated with Lactobacillus to start fermenting sugars in whatever you add it to.
Traditional ginger bugs don't use any 'ginger beer plant', but I assume GPB could be used as well. Ginger Beer Plant is already its own culture/colony. A ginger bug is a means to grow a similar culture without having to procure it from an already established colony. Two different things, but either may be used in recipes calling for a lactid acid starter.
Ginger bugs are more popular in the common kitchen than GBP because plain old ginger root is much easier to come by in a local grocery store.
A ginger bug does not provide enough ginger flavour for ginger ale. After you've grown a ginger bug, you must separately brew a batch of sweet ginger tea, enough tea to provide as much ginger ale as you wish to ferment, sweetened with enough sugar to further feed the fermentation.
To this ginger tea the strained ginger bug liquid is added. Generally, one cup of ginger bug per gallon of sweet tea is enough to do the trick. If you're left with extra ginger bug, you can leave the chopped ginger in it and refrigerate it in a sealed container.
Refrigerated, the culture will 'go to sleep' until you take it out, start feeding it sugar again, and leave it in a warm, dark place covered with a cloth to let it reawaken and turn fizzy again - adding some more water if you need the volume. Once it's fizzy again, you can use it to ferment a brand new batch of ginger tea into ginger ale.