  Assume that the vertices of a graph $G$ are always operational, but the edges
of $G$ are operational independently with probability $p \in[0,1]$. For fixed
vertices $s$ and $t$, the \emph{two-terminal reliability} of $G$ is the
probability that the operational subgraph contains an $(s,t)$-path, while the
\emph{all-terminal reliability} of $G$ is the probability that the operational
subgraph contains a spanning tree. Both reliabilities are polynomials in $p$,
and have very similar behaviour in many respects. However, unlike all-terminal
reliability, little is known about the roots of two-reliability polynomials. In
a variety of ways, we shall show that the nature and location of the roots of
two-terminal reliability polynomials have significantly different properties
than those held by roots of the all-terminal reliability.
