If you're traveling enough that gas mileage is a significant concern (i.e. a significant part of your budget), you probably would save money by using an efficient car and staying in motels, many of which have internet access available that's at least as good as what is found at most campgrounds...and often a good bit better.
Anyhow, some advantages of a truck camper: one less thing to register (in nearly every state), you can get into more out-of-the-way places particularly if your truck is four wheel drive, easier maneuvering in tight places generally. Some disadvantages: they're tight, and due to the limited space have limited capacities for water, propane, etc. Storage for stuff for full-time living is quite limited.
Some advantages of a trailer: more space, in some cases much much more space; you can easily leave the trailer at a camp site and take the tow vehicle to town or wherever you need to go; easier to climb in and out. Some disadvantages: you have a trailer to maintain; if it's not a fifth wheel, or if you're on the east or west coast, you cannot legally tow a second trailer; somewhat more effort to set up or break camp, in general.
Some advantages of a motorhome: Very easy to set up and break camp; you have ready access to the kitchen, bathroom, etc. while en route; you can usually tow something else if the need comes up. Disadvantages: chassis maintenance costs (tires, for instance--not too bad for most class C's, significant for a large diesel pusher motorhome); not great fuel mileage; you have to take your RV with you when driving into town or whatever, even if you aren't otherwise breaking camp, unless you tow a second vehicle.
For $15K, you might be able to get a (well) used but basically sound and fully functioning, probably class C, motorhome with a bit of shopping around; or a trailer in reasonable shape and a suitable tow vehicle. For the latter, it might be worth looking into full-size vans for the tow vehicle; many are (or can be) equipped to tow a pretty good sized trailer, and have adequate payload capacity to handle the tongue weight and also carry a useful load inside at the same time. A fair few pickup trucks and SUVs, especially the smaller ones, tend to have limited payload such that you can tow a trailer or carry some stuff, but not much of both at the same time.
As a general observation, pretty much any vehicle that can serve as one's house, with kitchen and bathroom facilities and so forth, is going to be pretty big and heavy, and hence neither inconspicuous nor fuel efficient.
