  We report the discovery of TOI-1444b, a 1.4-$R_\oplus$ super-Earth on a
0.47-day orbit around a Sun-like star discovered by {\it TESS}. Precise radial
velocities from Keck/HIRES confirmed the planet and constrained the mass to be
$3.87 \pm 0.71 M_\oplus$. The RV dataset also indicates a possible
non-transiting, 16-day planet ($11.8\pm2.9M_\oplus$). We report a tentative
detection of phase curve variation and secondary eclipse of TOI-1444b in the
{\it TESS} bandpass. TOI-1444b joins the growing sample of 17
ultra-short-period planets with well-measured masses and sizes, most of which
are compatible with an Earth-like composition. We take this opportunity to
examine the expanding sample of ultra-short-period planets ($<2R_\oplus$) and
contrast them with the newly discovered sub-day ultra-hot Neptunes
($>3R_\oplus$, $>2000F_\oplus$ TOI-849 b, LTT9779 b and K2-100). We find that
1) USPs have predominately Earth-like compositions with inferred iron core mass
fractions of 0.32$\pm$0.04; and have masses below the threshold of runaway
accretion ($\sim 10M_\oplus$), while ultra-hot Neptunes are above the threshold
and have H/He or other volatile envelope. 2) USPs are almost always found in
multi-planet system consistent with a secular interaction formation scenario;
ultra-hot Neptunes ($P_{\rm orb} \lesssim$1 day) tend to be ``lonely' similar
to longer-period hot Neptunes($P_{\rm orb}$1-10 days) and hot Jupiters. 3) USPs
occur around solar-metallicity stars while hot Neptunes prefer higher
metallicity hosts. 4) In all these respects, the ultra-hot Neptunes show more
resemblance to hot Jupiters than the smaller USP planets, although ultra-hot
Neptunes are rarer than both USP and hot Jupiters by 1-2 orders of magnitude.
