Foam-board has some rigidity to it, so it "stays where it is placed". The thickness of foam board also allows some multi-tined foam board "frog gig" type holders to be used. These have typcially four straiught metal tines welded to a strip of steel, and the profile look is somewhat like a capital letter " E ", but with four spikes sticking out from the E....those spikes are carefully driven into the foam board, and then the back of the gig, has a 1/4 x 20 threaded hole in it....that device can then be threaded onto gooseneck arms, which allows foam boards to be placed in to wacky positions in like, 3 seconds.
Foam boards also can be taped together, and then stood upright, in a V-shaped arrangement; paper does not have the rigidity to allow these types of uses. Paper actually WILL WORK as a reflecting medium though, as long as it can be positioned and secured properly. I have used paper sheets many,many times, especially for small fill-light duty.
Foam board is also fairly tough, fairly durable, and very resistant to tearing, and it can be easily held in basic positions using fairly crude, inexpensive "A-clamp" clamps, sold at home improvement stores all over the world, and just clamped to a light stand very rapidly and easily--the same would be pretty difficult to do with thinner, less-rigid paper sheets.
Thanks Trever1t and Derrel for the detailed explanations. For taking photos of metallic cylinder objects and metal plates, is it advisable to make an inverted |_| shape box and also the front cover (with a hole for the camera lens) using foam? The cylinder/spherical objects show reflections of me, the tripod, the ceiling and objects on both sides of them.