  Rapidly and differentially rotating compact stars are believed to be formed
in binary neutron star merger events, according to both numerical simulations
and the multi-messenger observation of GW170817. The lifetime and evolution of
such a differentially rotating star, is tightly related to the observations in
the post-merger phase. Various studies on the maximum mass of differentially
rotating neutron stars have been done in the past, most of which assume the
so-called $j$-const law as the rotation profile inside the star and consider
only neutron star equations of state. In this paper, we extend the studies to
strange star models, as well as to a new rotation profile model. Significant
differences are found between differentially rotating strange stars and neutron
stars, with both differential rotation laws. A moderate differential rotation
rate for neutron stars is found to be too large for strange stars, resulting in
a rapid drop in the maximum mass as the differential rotation degree is
increased further from $\hat{A}\sim2.0$, where $\hat{A}$ is a parameter
characterizing the differential rotation rate for $j$-const law. As a result
the maximum mass of a differentially rotating self-bound star drops below the
uniformly rotating mass shedding limit for a reasonable degree of differential
rotation. The continuous transition to the toroidal sequence is also found to
happen at a much smaller differential rotation rate and angular momentum than
for neutron stars. In spite of those differences, $\hat{A}$-insensitive
relation between the maximum mass for a given angular momentum is still found
to hold, even for the new differential rotation law. Astrophysical consequences
of these differences and how to distinguish between strange star and neutron
star models with future observations are also discussed.
