Why? Because!
Originally arched top and back guitars were made in imitation of the 'perfection' of the violin family. People didn't know why Strad's fiddles were so good (and we still don't), but if arching worked for them it ought to work on guitars, too. Which just goes to show how you can take an analogy too far.
Flat top guitars have a couple of weaknesses related to the way the strings work with the bridge. The basic function of the bridge is to tell the strings how long they are, so you have to have a way of stopping them at the saddle. The usual way is to 'break' the strings over the saddle at some angle, but that means either that the bridge has to be very high, so that you can get enough angle without them hitting the top, or that they have to tie into the top somehow. Flat top instruments usually do the latter, but this puts a lot of tension and compression in plane on the top from the string tension. In addition, since the strings have to be above the plane of the top there is a fairly large twisting force pulling the top of the bridge toward the neck. All of these forces on the top mean that you have to reinforce it somehow, and that tends to work against the production of sound; you'd really like to have the top be as light as possible, with only enough stiffness to discipline the vibration, in a sense. It's especially a problem in that the bending loads introduced by the bridge torque are precisely the sort of thing that wood does not resist well over the long term. All in all these problems tend to limit the size and longevity of flat top guitars.
With an arched top you can convert all of the loads into compression; the string is attached to a tailpiece hinging (somehow) off the tail block, so the neck and tailpiece compress the top along its length. The bridge also presses downward on the arch, which can resist this sort of force well over the long term so long as it is carefully designed and not too thin or under braced.
The obvious down sides of the arched top acoustically are that it is necessarily quite stiff at the bridge, so that it can resist the downbearing load, and that makes it harder to drive into vibration; it has an inherently high 'impedance'. It is also not possible for the top to respond as much to the twice-per-cycle tension changes of the string, which can at least effect the timbre of a flat top guitar, although we're not sure yet whether that's a _good_ thing.
A less obvious acoustic effect of arching is that it changes the effective 'damping factor' of the wood. Basically, carving an arched plate out of, say, a piece of maple, 'fools' it into acting like a piece of rosewood, at least at low frequencies. It will tend to respond well over a more narrow frequency range than a flat top, which tends to give a 'sharper' and more 'forward', if not 'harsh' tone. This may be one reason why rosewood archtops are not very common: too much of a good thing!
Another down side of carved arched plates is just the amount of work they take to make. You start with a thick slab of (expensive) wood, and throw 3/4 to 7/8 of it away. And not just _any_ 3/4 to 7/8. It appears that the exact shape and thickness distribution of the arches is critical to getting the best sound, and these things must be very carefully controlled. There are ways, ranging from a 'Lancelot' to the drill press planer to CNC, to reduce the dog work some, but in the end, in order to take the wood itself into account, there is still a lot of plain hand work involved with planes and scrapers.
At least you don't need to worry about back bracing on archtops: there is none. The top does need to be braced, more or less just to make up for the soundhole(s), and fitting those is another part of the 'fun' of making archtops.
All in all you should allot about three times as many hours to your first archtop as you do to a flat top instrument. There will be plenty of challenges; things that might not occur to you now. How, for example, will you cut the binding and purfling rabbets?
In the end there's no guitar quite so 'sexy' as a well designed and built archtop, though. Whether you make it simple and show off the nice lines and beautiful wood grain, or elect to make it an Art Deco tour-de-force of inlay with a great shaded 'burst finish, it can be as elegant and sophisticated as you can manage. A good one can even sound pretty nice!
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