They are mainly different in the ways they are manufactured
Expanded metal sheet
Expanded metal sheet is made by first creating multiple slits in the sheet, and then stretching the sheet. The stretching creates a unique diamond pattern opening with one of the strands protruding at a slight angle. These raised strands can be flattened later in the process if desired. As you can see this process creates no waste (thus keeping down production costs) and it can add structural strength to the product.
Perforated metal sheet
Perforated metal sheet is a product that is made from sheet steel that has been fed through a machine that punches out round holes (or other designs). These holes can be straight rows or staggered to increase the amount of the openings. Typically the perimeter of the sheet has a boundary where holes are not punched; this adds stability to the sheet. The removed metal from the holes can be recycled but it also increases the cost of the product. The larger the hole size (or increased amount of holes), the greater scrap volume, and therefore costs can be increased.
Wire mesh (welded)
Welded wire mesh is a metal wire screen that is made from various alloys including steel, stainless steel, brass and copper. It is available in various sizes and shapes. Grids of parallel longitudinal wires are welded to cross wires at the required spacing, using electric fusion. The machines that are used to produce the mesh have precise dimensional control.
Wire mesh (woven)
Also available in steel, stainless steel, brass and copper, woven wire mesh is made as a cloth with wire threads woven at right angles. Wires which run lengthwise are known as warp wires, whilst those running perpendicular are weft wires. There are two common styles of weave: plain weave and twill weave. It can be made from various alloys including stainless steel, brass and copper. Wire cloth can be woven to create a variety of opening sizes and wire diameters.
-----------Adapted from www.metalsupermarkets.com