Why do you need the intrinsic pathway when you have faster extrinsic pathway?
In short: The extrinsic cascade reacts to damages of the blood vessel, the intrinsic pathway reacts to damages of the walls. But as with all things, the long version is ways more complicated:

The extrinsic pathway is activated when you damage tissue and factor VII come in contact with tissue factor (also known as factor III). Tissue factor is located in the membrane of the cells and is only released when the cells are damaged. 
Then the cascade starts and delivers a very fast large amounts of active thrombin, which then converts fibrinogen to fibrin to build a block of the damaged blood vessel. The step from the thrombin is common between both pathways.

The intric pathway is also called the "contact pathway", which starts when the factor VII comes in contact with collagen or anionic surfaces. 
The main advantage of this pathway is the strong amplification of the initial signal from the intrinsic pathway by a factor of about 1000 fold. The intrinsic pathway is also co-activated by active thrombin (which stands at the end of the cascade) meaning it is also a powerfull feedback loop to make the initial reaction much stronger. This also overcomes weak activation which where the active thrombin will be inactivated by Tissue Factor Pathway Inhibitor.
The coagulation cascade has a pretty nice article in the Wikipedia.

Interestingly it is now believed, that the start of the intrinsic pathway plays no or almost no role in vivo. There are people which "suffer" from a deficiency in factor XII which is necessary to start the intrisic cascade. These people have absolutely no problems with their coagulation, so the first steps are not necessary (any more?). This article presents findings, that it stabilizes a thrombus:"The intrinsic pathway of coagulation is essential for thrombus stability in mice."
Here are some further articles about the intrinsic pathway:


In vivo roles of factor XII.
Factor XII: what does it contribute to our understanding of the
physiology and pathophysiology of hemostasis &amp; thrombosis.