It is possible for glitches to cause other resets. For example, they may fire an interrupt or interrupt a communication like I2C. Depending on your software, those false signals might cause a reset.
Presumably, however, the same software that works on a PIC has simply been moved to the new processor. A software-based cause would require the new processor to be more susceptible than a PIC, which I assume in unlikely. I have no experience with ST processors, though.
Since your new design works with battery power but not with the AC power supply, that suggests to me that it is related to the power supply, not software or board layout per se, unless that layout affects the power supply differently than it did with the PIC.
John