You actually mention two DIFFERENT parks in your title and post.
Indiana Dunes State Park was established in 1925 and is about 2000 acres with about 3 miles of beach/shoreline. The park is mostly beach, dunes and oak forest. Only about 1 mile of road which goes to beach parking and the campground (very nice campground, by the way). Only one entrance and they charge a fee.
Indiana Dunes National Lakeshore (the National Park) was dedicated in the 1970's (Tricia Nixon came to speak on behalf of her father's "Bringing Parks to the People" campaign). It is much larger at over 15,000 acres and 15 miles of beach/shoreline. It essentially runs from Gary to Michigan City, Indiana (except for the steel mills at Burns Harbor). There are many places you can drive in the National Lakeshore, including Chellberg Farm (historic farm), Bailly Homestead (first European settler in the area), West Beach, Pinhook Bog, Mount Baldy (moving dune), and various beach access points. Many, many roads that are all driveable by your vehicle because they are nearly all US or State Highways (US-2, US-20, State Highway 49, or county or town public roads. Also a couple of visitor centers to see. The new one is on SH 49 and is a joint visitor center for both the State Park and National Lakeshore. The old visitor center is on Kemil Road and is now an interpretive center. The National Lakeshore has a small campground not far from Kemil Road, but I don't know if it would accommodate a motor home. I used to work as a Ranger at the National Lakeshore a long time ago.
In short, the State Park is nicer for its unspoiled dunes, its pristine beach, its good campground and its many hiking trails. The National Park is nicer for its many and historic places to visit and drive to, and for Mount Baldy, the big moving dune. The national park offers more interpreters and programs than the state park.
Link to State Park MapLink to National Lakeshore Map