SiC and GaN address different sectors. GaN is an often superior replacement for Si superjunction MOSFETs in the ~600V segments (mains switchers / rectifiers). As an example you could read about Google's Little Box Challenge from 2015, which was dominated by teams using GaN technology. SiC is best for higher utility voltages that GaN can't (yet?) address. I believe SiC excells at very high powers due to its thermal properties, such as locomotive and replaces IGBTs and to some extent tyristors.
SiC are MOSFET or JFET with SiC instead of Si. These transistors are rather similar to traditional Si, but with better properties in many respects. I hope this image from Mitsubishi Electric is sort of correct in capturing the essential benefit of using a wider bandgap material:
Namely, you can reduce the drift layer thickness because the breakdown electric field is larger. That leads to a lower resistance per device area. This in turn allows you to make the transistor smaller laterally for a given power which reduces gate capacitance for the same power. This in turn allows higher frequency or higher efficiency. Another advantage is the lower reverse recovery parasitics for SiC but I don't know why.
GaN transistors are substantially different, being usually HEMTs and not MOSFETs. Nevertheless they are controlled with a gate voltage. Their gate charge is very low for the same power allowing very high frequencies while still offering rather high breakdown voltages, albeit not as high as SiC. They also conduct in reverse, mimicking the body diode aspect of traditional 3-terminal MOSFET, but have no reverse recovery, as the reverse and forward conduction path are identical.
Graphics like this show that the theoretical figure of merit is better for SiC than for Si, and better yet for GaN. But I don't know exactly how this is calculated. Presumably the logic is the same as in the image above. Higher breakdown voltage allows you to reduce the drift thickness. Factor in the drift layer resistance and you can calculate such a figure of merit. But this also skips over other important metrics, such as reverse recovery and device type. For example, the whole drift layer argument does not really apply to HEMTs. This graphic seems to indicate that either GaN is still far from mature or that a HEMT is inferior to a MOSFET for power switches, in case the material allows both.