An extraneous solution to a rational equation is traditionally one that will result in a hole. For example $\dfrac{x^2-4}{x-2}=0$ has the extraneous solution $x=2$ because it is a solution of the numerator but does not work in the original function (usually they are a little better-hidden). As a graph, this would look like a line $y=x+2$ with a hole at 2.
However, an asymptote is like $x=2$ in the equation $y = \dfrac{x^2+4}{x-2}$, which goes to $\pm\infty$ at $x=2$. You won't get an asymptote as a solution usually.
Edit: you can get an asymptote as a solution by the way.
$\dfrac{x^2-4}{(x-2)^2}=0$ has both an asymptote and an extraneous solution at $x=2$.