Durability - A double layer hammock can provide you a little more safety margin against abrasion damage or snags. In some cases, a tear or snag in a single layer hammock can mean the end of a trip and/or a cold night on the ground. A minor snag or tear in a double layer hammock can often be repaired as long as the damage is contained to only one of the two layers.
Insect Protection - Some people claim that a double layer hammock will provide more protection from insects getting to you from below your hammock. I'm not at all sure of this because I personally always use an underquilt or sleeping pad under me. Also, I treat my personal hammocks with permetherin to combat insects and I've never once had this issue.
Bottom Insulation - Sleeping pads can be terribly frustrating in a single layer because they're always trying to squirt out from under you. If you intend to use a sleeping pad, a double layer hammock works much better. If using an underquilt, a single layer hammock is usually sufficient.
Heavier layer on the outside or inside?
Definitely on the outside. This provides you the best durability against damage from brush, saplings, briers, etc. Also, this makes for a stronger hammock that will last longer if you are using a sleeping pad. Using a double layer hammock together with a sleeping pad of any substantial thickness places almost all the stress on the outside layer only until it stretches out and is "helped" by the inside layer. If the users weight greatly exceeds the capacity of the outside layer, damage or failure can easily occur.