I've noticed that the stores sell both regular (non-dimmable) and
Dimmable LED bulbs. The dimmable ones are more expensive. What is
different inside of them?

I dont use dimmers, or I'd try this.

What happens if a regular bulb is connected to a dimmer?

I'd assume a dimmable bulb will work fine on a circuit without a dimmer,
or is there a reason to NOT use them if no dimmer exists? (other than
the extra cost).