Capacitor
Using the larger capacitor should not be a problem. It appears that it is used along with the resistor and potentiometer to control the timing of the flash (frequency). When the capacitor is charged to a voltage that is sufficient to ionize the gas in the neon bulb a spark jumps turning on the SCR and allowing the capacitor to dump it's charge accross the trigger coil. The higher capacitance would only lead to a slightly longer charge time for a fixed resistance and slightly slower discharge through the coil. I suppose, in theory, this would mean less output voltage from the coil (not as large dv/dt at the primary). This should be negligible. In short, I do not see a way that the larger capacitor would damage the flash tube.
I'm not 100% sure about the answer for your next question. From my experience, anything labeled as a "trim pot" seems to be slightly more accurate (usually meaning more turns) and is typically adjusted with a tool or screwdriver. "Pot" seems to be a more general term and usually applies to variable resistors with a knob that can be adjusted by hand. This is the only difference as far as know.