We obtain two characterizations of an odd-dimensional unit sphere of dimension $>3$ by proving the following two results: (i) If a complete connected $\eta$-Einstein $K$-contact manifold $M$ of dimension $>3$ admits a conformal vector field $V$, then either $M$ is isometric to a unit sphere, or $V$ is an infinitesimal automorphism of $M$. (ii) If $V$ was a projective vector field in (i), then the same conclusions would hold, except in the first case, $M$ would be locally isometric to a unit sphere.