A fresh carbon/zinc primary battery has an "open circuit" voltage of 1.54 volts.
A fully charged NiCad rechargeable normally has an open circuit voltage of about 1.3 volts.
If your flash uses a typical 4 AA cells, the disposable batteries will feed it 6.16 volts when new, while a set of fully charged 4 NiCads will only show it 5.2 volts.
The lower voltage of the rechargeables is enough to prevent some devices from working with NiCad rechargeables.
There are other rechargeables that you can try if the common NiCads don't work in your device, but it can get pretty expensive doing it by trial and error.