There is a very direct relationship which answers your question, and I'll state it in the way I first learned about it (but you can derive a different connection by passing between dimensions):

**The 2-dimensional reduction of the Seiberg-Witten equations are the ***(abelian) vortex equations.*

The $SU(2)$-vortex equations on $\mathbb{R}^2$ are a Yang-Mills-Higgs equation, and is a 2-dimensional version of superconductivity, which is actually defined on $\mathbb{R}^3$ with $G=U(1)\subset SU(2)$. Here the YMH-equations are precisely the *Landau-Ginzburg equations *and $\phi$ represents a *Cooper pair* (a bound state of two electrons). Minimal solutions to this have $0=D_A\phi=d\phi+A\phi$ and hence $0=D_A^2=F_A$ which physically represents the *Meissner effect* (the expulsion of magnetic fields from the bulk of a superconductor).

You may have heard of "monopoles" in relation to SW-theory. That's because the 3-dimensional reduction of the Seiberg-Witten equations are the (abelian) *Bogolmony equations* which define monopoles. As above, $SU(2)$ vortices and monopoles are inherently related, and are dictated by an $SU(2)$ Yang-Mills-Higgs theory on $\mathbb{R}^n$ for $n=2,3$. The exact relations with everything I have mumbled will take more time to discuss (for instance, the 3-dimensional equations describing monopoles and also superconductors are slightly different, depending on the the existence of a potential and the type of representation (for the gauge group) you use).