Why choose delta over star connection for a 3 phase motor with a VFD?
For a given motor, the fundamental difference between the star and delta connections is that the rated voltage for the star connection is 1.731 X the rated voltage for the delta connection. That way a motor can be designed to run at either of two voltages, 240 V, delta or 415 V, star for example. If you connect the motor delta and connect it to 415 V, that is too much voltage for the design. The motor will draw excessive current and overheat. If you connect the voltage star and connect it to 240 volts, that is not enough voltage for the design. Since torque capability is proportional to the square of speed, the motor will only be able to produce about 33% of rated torque without running slow and drawing too much current. With a VFD the effect would be the same, the motor will produce less torque per amp of current.

For normal operation of a motor with a VFD, there is no choice, you must connect the motor for the voltage that matches the VFD voltage rating. The maximum VFD output voltage rating is normally the same as the input power voltage rating.

What could be done with a VFD is to use the delta motor connection with a 415 V VFD, but set the VFD to have an output of 240 V at 50 Hz and 415 V at 86 Hz. That would increase the power rating of the motor X 1.731 because the motor the nearly the same torque capability at a high speed. There would be some increase in motor losses, so the motor's torque capability is reduced somewhat. There would be some concern about bearing life and motor vibration, so it is not a good idea to operate a motor that way without checking with the motor manufacturer.