Well, some people do all one or the other. Each has pros and cons:
Nylon
Pros - Durable, easy to cut (melt)
Cons - No flex, more stress on connection points (possible damage to strapping/armor might occur)
Elastic
Pros - Flexible
Cons - Harder to work with (fray check, sewing usually required)
If you go with elastic I'd recommend getting the heavy duty thick stuff and doubling it up, personally I used nylon for most areas and only have elastic in a few spots for flex, for example one side of my ab/kidney connection is nylon but the other is elastic so I can actually get the snaps on and off.