Although plywood seems quite rigid, because it is wood, an organic material, it inherently moves -- not very much, but enough to cause damage to the tile installation. Plywood can expand and contract in three directions. According to the Ceramic Tile Institute of America, "A typical sheet of Douglas fir plywood unrestrained will expand and contract 7/32 inch over its length and over 3/32 inch on its width."
A sheet of plywood that is securely nailed or screwed to a frame may not move quite so much, but expansion and contraction are still possible. What's more, each individual layer in the plywood will expand and contract independently. Movement in the plywood will cause the tile grout to crack and tiles might even pop off.
A related problem with plywood is moisture. The first thing to see failure in a tile job mounted on plywood will be the grout. Although tile can form a strong bond to plywood, the wood will absorb ambient moisture and expand and contract so much that eventually the grout will crack. A Portland cement-type grout will crack and crumble, leaving grit on the tile; an epoxy-type grout will form small cracks around the tile.
When tiling rooms that naturally see a lot of moisture, such as bathrooms and kitchens, use a cement backerboard-type material as backing for wall tile. This is the same material recommended for use underneath tile floors in these rooms. Additionally, apply a vapor barrier fabric to the wall underneath. A grout sealing product applied over finished, dry grout is also a wise idea.