Chicken wire won't keep out any predators other than hawks. It's not strong enough to keep out raccoons.
Hardware cloth is wire that's been welded at the intersection of the vertical and horizontal wires. That makes it stronger and more secure. Chicken wire is woven, meaning that two strands of wire are just twisted around each other. When a raccoon puts its paws on chicken wire it can pull an opening large enough to get inside. Another factor is that chicken wire is usually a pretty high gauge (that's a measure of the thickness of the wire). The higher the gauge, the thinner and weaker the wire.
Openings in wire larger than 1/2" by 1/2" allow for the danger of reach through predation: that's when a critter can reach in, grab a bird, and pull pieces of it out through the wire, bit by bit. It's ghastly. To prevent this, some people who use low gauge wire that has larger openings will wrap the bottom 2 or 3 feet of their fence with hardware cloth or even something solid.
To deter digging predators, you don't have to dig a trench and bury wire. You can attach a welded wire apron to the baseboards of your run, extending outwards flat on the ground about 2 feet or so, and staked down with landscaping staples. When a critter tries to dig in, it begins digging at the edge of the vertical fence, hits the apron, and doesn't realize that it needs to back up and start digging beyond the edge of the apron. It's much easier to install an apron than it is to dig a trench and bury wire, and it works just as well if not better.
