I travel by RV and I travel by car and stay in motels.
For the motels I use when traveling alone, comparing motel rates and RV/Park campground rates, the RV saves me enough on motels to pay for enough gas to move it about 50 to 100 miles a day, one-night stays. So if I'm moving far and fast, the car (36 MPG and $40-90 a night) is a lot cheaper than the RV (8 MPG and $10-50 a night). If I'm staying for a while, several days at each stop, the RV can be cheaper.
For what you are talking about doing, the RV would probably save you money compared to traveling by car and staying in motels, especially if you are good at seeking out the low cost places to stay. Where I travel, I can make good use of municipal RV parks, and public campgrounds at Corps of Engineers flood control reservoirs. In other parts of the country, the less expensive places may be run by the National Park Service, Forest Service, or Bureau of Land Management.
What tends not to work so well is visiting major urban areas. To find a reasonable place to park a RV for a visit, you may find yourself commuting for your sightseeing, and sometimes missing the nightlife that is so much a part of the urban experience. However, staying in city center hotels to get that experience can be very expensive, not unusual to pay $200 or more for a "business" class accommodation.