Big Bend Ranch State Park is far more rugged than Big Bend National Park. It's a remote park where the majority of trails and attractions are in the interior. The interior is the main draw, and it attracts off-roaders. That's because the park is almost entirely off-road, except for the one paved road going straight through it: River Road (FM 170), also known as Camino del Rio. For the rest of the park, you'll need a 4x4 or a high-clearance vehicle.
I had a Toyota Camry and got a flat tire on the flat dirt road that serves as the park's main interior route. Once I get a lift kit and terrain tires installed, I'll have no problem, but on a serious note, you will not be able to explore everything this park has to offer without a vehicle capable of going off-road.
Big Bend National Park, on the other hand, has nice — I wouldn't say smooth, but paved — roads. With the Camry, it shook a little more and the roads felt a little rougher, but we had no problem driving all over the place. If you do happen to get a flat, we had to call out Diego Tire Repair Shop, who saved us from attempting Big Bend National Park on just a spare (and I probably would have done it, too). I'll link my crazy 800-mile spare story here once I write it. Big Bend National Park is doable with a normal everyday car and a will to survive the sun and heat. A big highlight of this park is the scenic driving; in fact, River Road continues through Big Bend National Park, but I'd say the section in Big Bend Ranch State Park is much better.
Driving-wise, especially on Ross Maxwell Scenic Drive, Big Bend National Park was more scenic with amazing views of the mountain ranges, while BBRSP had more limited views but was still very scenic. It just couldn't compete with the size and volume of the Big Bend National Park mountains. On the other hand, the drive through BBRSP was more fun, as it contained the steepest grade highway section in all of Texas, with twists and turns up and down, you didn't know which way the road would turn next. I took it a little faster than I should have. At points, the elevations and drops over the hills looked like you were driving off a cliff, you literally couldn't see the road. Then out of nowhere a crazy turn would appear, and I'd have to brake hard or risk flying off the road or into a cliff-side.