Generate a checksum/crc across the image so you can see if it is the same.
The debugger may enable clocks, fiddle with GPIO, or internal settings to meet its needs, your code might have dependencies on these, and if the processor runs or starts more slowly.
Dump peripheral registers, starting with RCC, to see if you can identify differences on/off the debugger.
Check BOOT0 pins.
Add general diagnostics/telemetry so you can understand the failure mode, or identify changes in behaviour and code flow.
