To understand the most significant difference, we must first understand how a conventional tokamak generates part of its magnetic field: the poloidal magnetic field is created by driving a toroidal current, i.e. a current along the (toroidal) plasma column. That current is driven by ramping up (or down) a voltage of a central solenoid. The conventional tokamak is more donut-shaped where the spherical tokamak (ST) is more similar to a cored apple and has thus very little space for a central solenoid. This means that the toroidal current to generate the poloidal magnetic field has to be driven by other means, e.g. radiofrequency heating.
The spherical tokamak has in general a lower background magnetic field than the conventional tokamak. The resulting plasma beta, the ratio of plasma pressure to magnetic field pressure, is higher in a ST. One could therefore say that you get more plasma per magnetic field unit in a ST which translates into a financially cheaper device (since magnetic field strength (or rather the volume) is one of the most costly part of the conventional tokamak).
In terms of stability, the ST behaves differently than a conventional one. Another point is the small distance of the coils to the plasma in the ST which is problematic in a reactor-like device where you have neutrons (from the fusion reactions) that need to be slowed down to shield the coils.
As for current research projects, there are the MAST-U device in the UK, NSTX-U in the US, QUEST in Japan and the ST line of the company Tokamak Energy.