As above, the photoflash cap will have a lower equivalent series resistance, which means 2 things the cap will produce a higher instantaneous current meaning a shorter brighter flash and also will not get as warm, much like a thicker cable carrying a high current.
If you fitted a standard cap it'd work, but the flash wouldnt be as bright, and it go on a ms longer or so, and if you took pic after pic the cap would heat up and if a high power flash might even explode.
Elecro caps in pc power supplys have a low esr, but still not low enough.
not only must they have low ESR, but have very robust connections between the internal foil plates and the terminals
Interestingly, low ESR doesn't help a photoflash capacitor stay cool. Image a charged electrolytic capacitor, discharging into a short (or very low resistance). The discharge current magnitude and waveform will be determined by the capacitor's ESR and ESL (mainly determined by the ESR).
Now, consider a capacitor with 1/2 the ESR (and the same ESL). The discharge current will be essentially doubled in magnitude and speeded up by a factor of 2. The instantaneous dissipation in the ESR will be increased by a factor of four, the loss per discharge increased by a factor of two (not by a factor of 4 because the effective discharge time is half as long), and the capacitor will got hotter, not cooler.
This is a result of the discharge current being determined by the parameters of the capacitor itself, and not by the external circuit parameters. That is what happens when discharging the capacitor into a short (or near short). The external circuit will, of course, have
some effect on the discharge, but when discharging into a near short, the capacitor's own parameters will dominate the discharge characteristic.
For certain applications, for example a defibrillator, an external inductor is used to shape the discharge, and then a lower ESR may be beneficial. But for typical photoflash use, external inductors are not used.
Here's a list of ordinary capacitors. Look up the 350 Volt, 390 μF capacitor specs. The rated ESR is .51 ohms:
https://www.electro-tech-online.com/custompdfs/2013/07/381-383.pdf
Here's another list. These capacitors are specifically designed for photoflash service. Look up the 360 volt, 400 μF capacitor specs. The rated ESR is .497 ohms, essentially the same as a non-photoflash capacitor:
https://www.electro-tech-online.com/custompdfs/2013/07/7P.pdf
Somewhere I have an old application note concerning energy discharge capacitors. The ESR of electrolytic photoflash capacitors is not as important as the really good electrical connections to the foil, making the foil thicker, and getting the heat out. Regular electrolytics come in only a few case sizes, and many of the standard values are packaged in the next larger can, so that there is substantial space around the wound foil; in other words, the wound foil doesn't fit tightly into the can. Photoflash capacitors are sized so the wound aluminum foil fits tightly into the can, enhancing heat flow to ambient.