I'll admit to a game prejudice. I prefer mounted boards.
This may be because my formative years were spent with Avalon Hill games. I came across SPI and some of the other paper map companies a little later on.
One welcome trend I've been seeing is the proliferation of mounted map boards -- often as upgrades for games that originally came out with paper or cardboard maps. Some recent examples include Commands & Colors Ancients, Paths of Glory, Twilight Struggle, The Awful Green Things From Outer Space and Shiloh (Columbia Games).
The debate between mounted map fans and the unmounted aficionados is one that can't really be settled, one way or the other. Partisans of each can point to various reasons why their favorite display is better.
Obviously the biggest advantage of paper or cardboard maps is expense, followed closely by weight and space. A paper map game costs less to make, costs less to ship and, all things being equal, will take up less space in storage. For some purposes paper maps are indispensable -- magazines with games in them, for example, would be impractical with mounted boards for reasons of postage alone. Many players like to overlay Plexiglas on their maps to preserve the surface, smooth out wrinkles and hold the map in place. All valid points, although the reason why you need to smooth out wrinkles and hold the map down is because it's NOT mounted. In some games it's convenient to be able to write on the playing surface, which is easy to to with Plexiglas. Some players also like to use magnetized counter holders which is easy to do with a papper map placed on a metal sheet. They can be mounted on a wall or stored in drawers (specially made or map drawers) between plays. Paper maps and cardboard maps also avoid problems where map cuts meet. There can be a seamless transition between two map sections separated by a cut, whereas a mounted mapborad often is forced to leave a small gap.
Despite all these factors, when given a choice I'll opt for a mounted board. The main advantages I see in mounted boards is durability and providing a steady playing surface. Durability comes in both short-term and long-term ways. For the short-term, mounted map boards are usually more resistant to accidents such as spills and tears. For the long-term, if properly stored, mounted map boards can last decades. I have a 1961 edition of Avalon Hill's Civil War wargame which appears practically new -- despite being 50 years old! In contrast, long-term storage of a paper map is brutal and I have several games that have seen little table time that have maps splitting at the folds. A mounted map is less likely to be disturbed during a game than an unsecured paper or cardboard map. Securing the paper map usually means either taping it down or that trusty bit of Plexglas. Plexiglas isn't as portable as a folded mounted mapboard, however, a factor if you're not playing at home. A mounted mapboard can allow a game to be played on a less than ideal surface such as a picnic table, rough wooden table or even a blanket or the bed. Finally, and very subjectively, there's an aura of quality about a mounted board that makes the playing experience just a little better.
I won't thumb my nose at an game just because it has an unmounted mapboard. I have no shortage of unmounted map games in my collection. Some of my favorite manufacturers almost never use mounted boards, such as Columbia Games and MMP. But finding out a game has a mounted mapboard is definitely a selling point for me and I'm willing to lay out a little extra dough for one -- or buy one separately. I'll probably get the mounted For the People map, for example, just like I picked up the mounted maps for Commands & Colors Ancients when they became available.