Three different factors at play when you compare chicken wire (sometimes referred to as hex wire) to welded wire (sometimes referred to as hardware cloth). The first factor is the thickness of the wire, or gauge. The higher the gauge number, the thinner the wire. Thus 20 gauge wire (common for chicken wire) is thinner than 19 gauge, etc. 20 gauge wire is quite thin and easily bendable in your fingers. 16 gauge wire is not easily bendable in your fingers, and really gives you a workout when you're cutting it by hand with wire cutters.
Another important factor is structure. With chicken wire, the strands of wire are simply twisted around each other, which means that the openings can be stretched when force is applied. With welded wire, the connections are (obviously) welded, which is much stronger. In order to break down the structure of a welded fence, enough force must be applied to break the weld.
The other factor to consider is the size of the openings between the wires. With chicken wire, the openings between the wires are usually 1 inch or more, which means critters can easily poke their paws through (as well as large mice/rats and snakes can just go right through). It's very difficult to find chicken wire with smaller openings. On the other hand, welded wire comes in a range of wire spacings, from as small as 1/4" by 1/4" all the way up to 2" by 4" and larger. Wire with the very small openings is often higher gauge wire than the other kinds, though.
Unless you buy stainless steel wire (which is Yikes!!! expensive), you'll be buying a galvanized wire that has a zinc coating to deter rust. There are two kinds of galvanizations for welded wire: galvanized before weld (GBW) and galvanized after weld (GAV). The GAV wire is usually more expensive and it's considered better (stronger) than the GBW. GBW mesh can begin to rust at the connections because the welding process removes some of the zinc coating from the wires.
Finally, you can buy vinyl coated wire that may add a little extra life to the product, but the wire underneath should be galvanized, also. I buy the vinyl coated hardware cloth for my runs mostly because it looks better to me than the plain wire, and I'm too lazy to paint all that wire black.
