Insulated gate bipolor transistor is basically an electronic switch used for high efficiency switchingIBGT is a switching transistor.
The Igbts which are known to be the most reliable are the German Siemens InfinionsI believe IGBTs require more complicated control circuitry than MOSFETs, with optocouplers, etc. This could make manufacturers choose MOSFETs.
Top manufacturers started using IGBTs and advertising this, implying it's significantly better (I'm not saying it's not better, I have no idea) so Chinese manufacturers followed, making people think "ooh, it uses IGBTs, just like Miller, I'll get that".
Much better explanation that we all managedIGBTs and MOSFETs are very similar devices, and operate (superficially) in pretty much the same way.
MOSFETs are better at higher frequency, but because they're not quite so good with high currents you generally need more to equal an IGBT, so can be more expensive to implement.
Contrary to popular belief, manufacturers don't choose to change a thing 'because better', it's cost based. Always has been and always will be. The only time that changes is if it could seriously damage a reputation, and then only if they care about such things.
Designed properly, either will make a good welding supply