Is this termite or moisture damage

Bring in an engineer. The termite inspector's job is to tell you that there is damage and where. The engineer's job is to tell you how that affects the building and what can be done about it. (A home inspector likewise is generally tasked with pointing out what's an issue rather than telling you how big an issue it is, though you can sometimes read between the lines, especially if you walked through the inspection with him.)

Damage that affects structural members of a house may not be repairable. Folks in my neighborhood had to demolish an old farmhouse they'd bought because it had a combination of problems (not termite-related), and nobody was willing to try to fix either one for fear of causing the other to fail catastrophically.

(The termite inspector may give you a general sense of whether the damage is cosmetic, minor, or serious... which will at least tell you whether you do need the engineer's opinion.)

You said both contractors "think" you have termites. They should know and be able to point out evidence like Tester101 pointed out. Both prices seem high for just basically spraying, especially since you don't have a slab that has to be drilled through. Near the front door, they could have just came in under the door. If they are subterranean, their should be evidence of how they got in - typically mud trails coming up from dirt, up the piers and into the house. They can also look for any things that are conducive to attracting them, such as any boards (pressure treated or not) that are in contact with both the house and the dirt. Just because you have seen a few doesn't mean they are already in your walls. A less expensive spraying might take care of it. I would call another guy and see what he thinks.