  The inclination distribution of the Kuiper belt provides unique constraints
on its origin and dynamical evolution, motivating vertically resolved
observations of extrasolar planetesimal belts. We present ALMA observations of
millimeter emission in the near edge-on planetesimal belt around $\beta$
Pictoris, finding that the vertical distribution is significantly better
described by the sum of two Gaussians compared to a single Gaussian. This
indicates that, as for the Kuiper belt, the inclination distribution of $\beta$
Pic's belt is better described by the sum of dynamically hot and cold
populations rather than a single component. The hot and cold populations have
RMS inclinations of 8.9$^{+0.7}_{-0.5}$ and 1.1$^{+0.5}_{-0.5}$ degrees. We
also report that an axisymmetric belt model provides a good fit to new and
archival ALMA visibilities, and confirm that the midplane is misaligned with
respect to $\beta$ Pic b's orbital plane. However, we find no significant
evidence for either the inner disk tilt observed in scattered light and CO
emission or the South-West/North-East (SW/NE) asymmetry previously reported for
millimeter emission. Finally, we consider the origin of the belt's inclination
distribution. Secular perturbations from $\beta$ Pic b are unlikely to provide
sufficient dynamical heating to explain the hot population throughout the
belt's radial extent, and viscous stirring from large bodies within the belt
alone cannot reproduce the two populations observed. This argues for an
alternative or additional scenario, such as planetesimals being born with high
inclinations, or the presence of a `$\beta$ Pic c' planet, potentially
migrating outwards near the belt's inner edge.
