Continuous lights are nice because they allow for you to see the work you are doing, but are also significantly more costly for the light they produce. The light that meets your need is the light to use regardless of strobe or constant output. If you need to do precise light placement and power isn't a concern, then continuous is a great option. If you need something more affordable or that doesn't take as much power, flashes work great too.
As far as the various reflectors and diffusers, they all serve similar purposes. The more direct a light source, the harsher the edges and the highlights, the more spread out the light source is, the softer it becomes. Umbrellas, softboxes, beauty dishes and reflectors (of which umbrellas and beauty dishes are often a type unless using as shoot through umbrella) serve to spread the light out.
Softboxes are more direct since they are transmitting light instead of reflecting it, thus they'll tend to be stronger near the middle.
Umbrellas reflect the light and try to spread it out very evenly.
Reflectors in the room itself are designed to reflect light sent towards them such that it fills in shadows without an additional light present.
Honeycomb grids limit the direction in which the light can travel, thus focusing it in a particular direction and can be used to help with selective lighting.
Similarly, barn doors block light from going in an unintended direction and allow further refinement of the area being lit.
How you use them all together is really a much deeper question than can be addressed in a single question, but hopefully this basic introduction gives an idea of the purpose of each of the devices and how they can be used in lighting a scene.