I'll answer the "When should I" of your question simply, and from the experience of someone with a music ed degree, but no formal voice training. I do know the basics of vocal training and education, and have had a little myself. I won't delve into the different physiological differences with the falsetto voice, since that's been awhile… :)
For most male singers, with enough training, they find they have a chest voice, a head voice, and a falsetto. I'm simplifying, but most discover these three ranges. The chest voice is the one you describe, it has a deeper resonance, and you feel many of the vibrations as they are happening. The head voice is a lighter sound and feel… it's tough to describe, but in my opinion it sounds a little closer to the resonance of the falsetto, and the singer's feel of their own vibrations decreases slightly HOWEVER they can feel a difference between that and falsetto.
Most singers will train all three parts of their voice extensively, and extend the ranges of all three, so that they can overlap and make seamless transitions. Falsetto, once strengthened, can be used to extend the range of the head voice when singing lighter repertoire. I've also noticed that strengthening the falsetto improves the chest and head voice. For example I've worked on falsetto enough that I can impersonate an operatic soprano. Not perfectly, by any means, but I can approximate the resonance and tonal qualities of, well, probably more like a mezzo-soprano. By doing this, I improved the resonance and consistency of the lower parts of my voice.
I'm not sure how much you know of the history of western (classical) music, but there used to be men called castrati, who were… modified… before they reached puberty, and so retained the high voices of boys, but the size and breathing capacity of full-grown men. After this process was finally abolished, some men became falsettists, or countertenors, often singing roles created for castrati. These men are just like you and I, except they choose to work and strengthen their falsetto rather than their chest and head voice. The point I'm trying to make is that these men are usually baritones, as something with the baritone voice usually results in a higher, richer falsetto.
So, depending on what your goals as a singer are… strengthen and work on your falsetto, if not to use it, at least to improve the lower ranges of your voice. I would NOT "shape" your throat in any abnormal way to modify the tone of your falsetto. Consult a voice teacher, at least for a lesson a month to keep you on track, and be up front with them about what you want to accomplish. The goal as a singer is to be stress-free; singing should be as effortless as talking.
However, for now, you can use the falsetto as an embellishment in songs you sing, use it to more easily reach higher pitches you can't sing in your chest voice, or just sing with the sopranos all the time. :)
If you listen to any musical theater music, there are good examples of use of falsetto. It's often used to dramatize a single outlier note, or even during a soft passage, to further soften the tone.
If you listen to folk music, it pops up now and then in there, sort of as an extension of the voice, but sparingly used to change the character.
However you'll most likely find that you will be unsure of what's falsetto and what's head voice when you listen to a softer song, especially in the musical theatre or classical realms. THIS is good falsetto.
Some people overlook it, and see it as a guy's circus sideshow… but all it really is is another portion of your voice that can be trained and improved just like any other.
Sorry for the overly long and rambling answer; you said you're an intelligent sort, so I threw you everything but the science, which percusse already gave you.