A group is a mathematical "thing", SU(2) is the set of all unitary 2x2 matrices with determinant 1. So there is the same group describing both spin 1/2 and the gauge properties of the weak interactions. But there is no physical connection.
Another example is how we use SU(3), we use it as gauge group for the strong interactions (SU(3)_colour) and as an (approximative) symmetry of the strong interactions of the three lightest quarks (SU(3)_isospin)
In the strong interaction, a quark has three colours so the spinor is (q_red, q_blue, q_green), thus this spinor transform in colour space and from the requirement that the Lagrangian should be invariant under this transformation, we can derive the gluons and their interactions with themselves and the quarks by their colour charge. So we can say that the strong force is independent of colour, we can transform the quarks to any colour we want and still have the same end result.
In the flavour symmetry, we say that the force between quarks are flavour independent, i.e it does not matter what kinds of quarks that participate in the interaction. up-down is same as strange-up etc. Thus the spinor is (u,d,s), and by imposing this SU(3) symmetry, we end up with the light hadron spectrum.
As you can see, SU(3) is used for two different things. i) The transformations of what colour quarks have, and ii) the transformation on what flavour quarks have.
Now this second symmetry is first of all approximative, due to the quark mass differences. And more, there are more than 3 quarks. So this is just an approximative, but illuminating, symmetry.
So now the SU(2) case, the SU(2)_spin is rotation of a 2-spinor in spin-space, and SU(2)_weak is rotations in weak-isospin space. The spaces are different, but works in a similar manner.
I can describe the number of CD-records I have, and the number of books I have with the same mathematical tool (arithmetic), but that does not say that there is a connection between books and cd's. The mathematical tool can be used for anything that I expect it to be useful for. Same with groups.
