Given my vague non-musician’s understanding of how a piano works, thie orientation of the strings doesn’t seem like it would be that big a deal. Yet no pro would ever play an upright. Why not?
The sound is richer and fuller, mostly because there’s more mass involved in the instrument itself to act as a sounding board.
An upright has a maximum height of 5-6 ft. Even with the strings diagonal, that’s a max length of 8ft or so, not ideal for the lowest frequencies needed. A grand can obviously provide a much greater length than this, and therefore the length of the longest strings is not compromised (the shape of a grand comes from the higher notes using shorter strings). This also gives much greater flexibility for the soundboard (under the strings, in a grand) to be shaped exactly right for the optimum acoustic properties. And an almost-simplistic point - the design of a grand allows the lid to deflect the sound into the audience, impossible with an upright. (AFAIK, the internal mechanism of a grand is better, but I’ll let someone who knows tell us about that)
Depends on the type of music. Try to find a honky tonk musican playing a grand.
Well, not just the mass of the instrument, the actual sounding board is larger, and the strings are longer, too, which gives a richer, deeper sound. A full-length concert grand is over nine feet long, which would be tricky to arrange vertically, although it has been done.
I see on preview that GorillaMan has made some of the additional points I was going to make, so I’ll just mention that a major advantage of a grand is orientation of the action–the mechanism that converts the player’s striking of the keys to the hammers hitting the strings. In a grand, the hammers hit straight up against the strings, and gravity pulls them back away. In an upright, a mechanism must pull them back to keep them from damping against the strings. Thus the grand’s action is much more sensitive and allows for much more rapid striking of the same note.
Plus it’s a lot easier to sit sexily on top of a grand 
I know nothing about music, but something about math.
Why is a piano (i.e. harp) shaped like that? That is, why do the string lengths form a complex, variable curve instead of just a straight sloping line?
Each octave higher (ie the same note, but higher) has doible the frequency. That means the relevant string needs to vibrate twice as fast. The ways to do this are to shorten the string, or to tighten it, or (in the case of a grand piano) both. This explains the concave part of the piano’s shape (the bit the singers lean against ). As for the rest of the curviness - it’s mainly because the strings are strung across an iron frame (a couple of hundred metal wires at high tension are strong), and this is the shape it’s easiest to make a strong frame in. Plus the general aesthetic niceties of a curve.
As GorillaMan pointed out, the physics of musical tones requires a doubling in length for each deeper octave. Look at a church organ sometime and you will notice the buttons (or ‘stops’) labeled 1 foot, 2 feet, 4 feet, 8 feet, 16 feet, 32 feet, etc. (There are also intermediate lengths, like 5 1/3, but never mind that for now.) When you double the length of the pipe (or piano string) you make the sound one octave lower.
The shape of a grand piano comes from the arrangement of these successively doubling string inside the cabinet. However, they are not arranged in a single straight line, nor are they all simple straight strings.
Hold your hands out in front of you at arms’ length. Keep one arm straight and lay the other wrist atop the first. This is roughly the same way the strings in an upright piano are laid. The high-pitched strings in the treble end are straight strings, two strings or three strings per note. The bass strings on the left are often coiled strings and are laid underneath the others, two strings per note—or only one string, at the extreme lower end.
Now, the OP—the reason to play a grand piano instead of an upright is that the grand simply creates more sound. The sounding board is an exquisitely shaped cone made of hardwood and it reflects the tones of the strings and amplifies them, in much the same way as a megaphone or a speaker cone would.
An upright piano focuses this sound primarily backward, away from the musician. A grand piano focuses this sound into the floor and, with the lid up, into the air above the piano as well. A grand piano isn’t always “better” if you have a very small room to play it in, because its volume will be overpowering.
Upright pianos don’t have the full functionality of the pedals, either.
A grand has the damper pedal, which lifts the dampers off the strings to let the strings ring continuously when you hit the keys; the una corda, which relocates the hammers so that they only strike one of the two parallel strings per note in the low midrange and one of the three parallel strings per note of the midrange-to-treble, so as to create a softer, more tremulous sound; and the sostenuto, which holds any dampers currently lifted off the strings (due to the note being depressed) so that they don’t descend back down onto the strings, letting those notes continue to ring while other notes subsequently struck are muffled with dampers in the normal way, so that you can sustain some notes while others are not sustained.
On an upright, the damper pedal works the same way as it does on a grand; but the una corda works differently, generally bringing the array of hammers closer to the strings to reduce their travel distance rather than relocating them to hit fewer strings; and the sostenuto generally doesn’t do anything at all. (On a few rare uprights it lifts the dampers from all of the bass strings while leaving the rest of the dampers unaffected, which sort of imitates the grand’s behavior if the notes you want to sustain are bass notes, and if you don’t mind the muddy blur caused by having all the other bass notes undampened).
I’m afraid you’ve been misinformed. The sounding board is essentially a flat, albeit very carefully shaped, piece of wood. Although its top surface is curved, there is nothing cone-like about it. Here’s a picture of a sounding board.
Furthermore, a sounding board does not literally function in the same way megaphones or speaker cones do. They are analogous, in that they all amplify sound, but a megaphone works by focusing and directing air pressure waves from a vibrating source (your larynx). A loudspeaker cone is itself a vibrating source of sound that functions more like the larynx than a megaphone. And a sounding board amplifies the sound of the vibrating strings by creating sympathetic vibrations in a much large mass. If you’ve ever used a tuning fork, you know its sound is quite soft if it is just held in the air. But if you place its foot against a table or other solid object, the sound is much louder because the object is made to vibrate as well. This is the principle of a sounding board.
Finally, the difference between the sound of a grand and of an upright is not merely volume. Grands typically have a much richer and fuller sound. And the other advantages I mentioned in my earlier post.
(How do I know all this? I did a lot of research before buying this piano.)
Sorry, commasense. All I used to do is sell them.
No, it’s not shaped like a traffic cone, but the sounding board does have a distinct crown, not flat as you stated. And the analogy is clear to megaphones and speaker cones; the sounding board provides a greater surface area of vibration than could be provided by the string alone. This can be read on the Steinway & Sons site regarding sounding boards. I’m at work and I can’t google for a cite at the moment, but I believe we are essentially describing the same thing.
The salesman in me just resisted getting too technical. 
Ah, that explains it.
I, too, said the sounding board was curved, but couldn’t allow readers to leave this thread with images of megaphones or speaker cones inside pianos that your description implied. And, as I mentioned, although there are rough functional analogies between a sounding board, a megaphone, and a speaker cone, each actually works on an entirely different physical principle.