Electronic – What’s the actual difference between edge sensitive and level sensitive interrupts

First, using abbreviations, especially ones that can have multiple expansions and on a international list is not a good idea. At first I thought "RC" meant resistor capacitor, but maybe you mean remote control instead? Do it right and spell it out, or at least define unusual or ambiguous abbreviations before using them.

Second, you need to read the PIC datasheet. After doing that you will see that this PIC has a number of external interrupts. The CCP module in capture mode can be one source of external interrupt, but there are others. These include the INT pin, the interrupt on change pins, and even the two comparators.

If you plan to use the CCP module as a PWM generator, then you can't use it in capture mode to generate a external interrupt. However, there may be other ways to achieve what you want other using a CCP module. Explain exactly what this "PPM" (another ambiguous abbreviation) signal looks like, what data is encoded by it, and how it is encoded. With that information perhaps we can get clever about interpreting it using some other hardware.

All that said, I see little point is deciding to use such a old PIC, especially if it doesn't really have the hardware you want. There are many many PICs to choose from. Step back and give the real specs of what comes in and what needs to go out of this block, and we can probably help pick a PIC and advise how to use it effectively. For example, some of the 24 bit core parts have a relatively large number of modules that can do CCP-like things. On those, there are separate modules for capture and PWM so they are not called CCP modules.

Added

You now say this block needs to interpret two usual "hobby servo" 1-2 ms pulse signals and produce two bi-directional PWM motor drive outputs accordingly.

The PIC 16F628A is the wrong PIC for this job. Since this is apparently a one-off, there is no point trying to save a $ or two at considerable cost in firmware complexity. Get the right PIC with the right mix of peripherals and have those peripherals do most of the work.

The dsPIC33FJ128MC802 looks like a good candidate after a couple of minutes with Microchip's MAPS parts selector tool. It comes in a 28 pin package, and has both input capture modules and PWM modules. It also has up to 8 motor control PWM outputs, which could allow you to use relatively simple H bridge hardware with the PIC driving each corner of the H bridge from a separate pin.

Level triggered (high or low) can allow the source to say "nevermind" or to keep the trigger active until the ISR gets around to it. Interrupt latency is not guaranteed on a single core with multiple triggers, though it's usually pretty fast. Generally, the signal for a level-triggered interrupt is itself edge-triggered and you have to clear it in the ISR or else you'll come right back into it again.

As Ignacio said, level triggered can also do something continuously while active, though you should write your software to not get stuck in an "interrupt loop". Not getting to your main code can be somewhat difficult to debug.

Edge triggered is good for things that happen once on some event. If the event happens again, then your response will happen again, so you'll need to be careful about repeated events like switch bounce.