Although they may look similar, SUVs and crossovers are two distinct types of vehicles. The defining difference is their DNA: an SUV is built on a truck platform, whereas a crossover is based on a passenger car.
With their rugged truck frames and running gear, SUVs are able to haul heavy loads and tow a boat or camper without breaking a sweat. They can also take you on off-road adventures (some better than others) and sail through deep snow and mud. Of course, this capability comes at price. SUVs are bulky, heavy, and drink a lot of gas. Maneuvering in urban traffic and parking lots can be a challenge, too much of one for many drivers.
Like cars, crossovers feature unibody (single-body) construction instead of a frame. This makes them lighter, more agile, and easier on gas. In short, more like a car. You get the same interior space and features as an SUV, but without the dynamic characteristics of a truck. The tradeoff is easy to predict: reduced capability. Most crossovers can tow and haul when the need arises, but at nowhere near the weight capacity of an SUV. It's the same story with leaving the pavement. A crossover with all-wheel drive can handle dirt roads and grassy campsites just fine, but that's where it ends. For true go-anywhere freedom, you need an SUV.
Beyond those practical considerations, some buyers just prefer the rugged personality of an SUV. If you've driven trucks all your life, you probably fall into that category. Even if you don't intend to bash boulders or tow an Airstream, you'll feel right at home in an SUV. On the other hand, if you're a car person at heart simply looking for more space and versatility, a crossover should fit the bill nicely.