Other than the physics (chemical storage vs. electric-field storage) one huge difference is they have different charge-vs-voltage curves.

The voltage across a capacitor is proportional to how much charge has flowed through it (it's the integral of the current). This simple mathematical relationship is vital for all sorts of analogue circuitry (and all circuitry is analogue eventually).

A battery, on the other hand, has a relatively constant voltage for most of its lifetime. A common 1.5v alkaline cell will only gradually droop to 1v or so before suddenly dropping off. NiMH batteries are very flat around 1.2v for most of their discharge. The voltage is determined by their chemistry, and is strongly affected by all sorts of real-world stuff like temperature and diffusion rates. For rechargeable batteries the relationship between state-of-charge and terminal voltage can be quite complex.

Of course real-world capacitors do have a lot of non-ideal behavior, like leakage, ESR, dielectric absorption, and microphonics, but even with all that they're a lot closer to a simple "voltage × capacitance = summed charge" relationship than a battery is.

There isn't going to be an abrupt cutoff between sound and not-sound, of course. But there'll be a range of pressures where it stops being as useful to think of the phenomenon as "sound" and starts being more useful to think of it as a vibrating thing occasionally imparting more/less energy to individual gas molecules.

My semi-educated guess is that's around when the mean free path of the gas molecules stops being small compared to the wavelength of the sound. But that's just a guess.

Here's an article that goes into the history of vaccine sites a little. TLDR is that injecting into a muscle gives a localized area for the vaccine and its adjuvant to sit and cause an immune response, but not so concentrated that you get scarring and such. However, that article notes we're not actually entirely sure why intra-muscular injection seems to work better or even if it's really that much better than hitting a vein. A longer article on the subject.

Muscles have a lot of blood flow, so the effect does spread through your body eventually.

Oral vaccines are tricky because your stomach/gut are pretty good at breaking down and neutralizing stuff we eat. (Not 100% perfect at it obviously.) Some vaccines are taken orally: there are oral vaccines for polio, typhoid, and cholera (and some others). You'll note that those are all also diseases that are transmitted by consuming contaminated water - so in this case it also helps that the tissue the vaccine touches (the gut lining) is also the first tissue the disease touches. That bonus doesn't apply to infections you get via the lungs though.

Your arm is sore because of the immune reaction - it causes minor inflammation.

Some people who have connective-tissue problems (joint hypermobility) also have IBS, and the researchers think maybe their IBS is caused by the connective body parts that holds their guts in position is not working right, so their guts get higgledy piggledy and feel bad.