Same issue with cars. In a nutshell, you can't drive both wheels equally with a solid axle because when you turn, the inner wheel will be rotating less than the outer wheel even in a very slight turn. Cars have differentials, too expensive for generic trikes.
Handling and performance suffers for everyday use when you have a solid axle with no differential action allowed. If you have a welded rear axle, you will feel like you are plowing. The bike will seem to skip and stutter as you make harder turns. Not very pleasant.
For racing though, it is desired to have a solid drive axle. Most trike racers will lean the trikes so to be only on 2 wheels when they turn which mitigates the negative impacts of a solid rear axle.