A short answer: The different Weil cohomology theories do not only provide cohomology groups or rings but also come with extra structure; e.g. l-adic cohomology comes with an action from a Galois group and de Rham cohomology comes with a Hodge structure (and Hodge structures can also be expressed as being an action from some group). This extra structure varies with the cohomology theory.
The motivic fundamental group should unify these extra structures -- they all should be shadows of an action of the motivic fundamental group.
For a start see e.g. the "motivic Galois group" section on the wikipedia page here and these notes by Sujatha Ramdorai. The book by Yves Andre referenced there is maybe a good next step.
