I can probably skip ahead here and just say what we might be all thinking: there is no difference! Who cares?! There might a clear division between SUVs and crossovers in the past but now the lines are blurred beyond repair.
It's worth repeating that SUV is an acronym for Sport Utility Vehicle. Crossover is a blanket term for a vehicle that straddles the line between two but has been co-opted to mean one that's stuck between a normal car and an SUV. In the early days, automakers resorted to calling them 'compact SUVs' to get around the unfamiliarity.
I'd guess that 90 percent of new SUVs/crossovers sold today are so labelled based on a decision from the marketing department on which designation would sell the most units, having nothing to do with its construction, platform, utilitarian attributes, pricing, or even size. If it's convenient to call it a crossover, that's what it is. Same with SUVs, and this can vary by region.
A great example here is Mitsubishi. Quite helpfully, their North American arm went through the trouble of explaining some of the key differences, and they all make sense. Unfortunately, while they classify the Outlander as a crossover, Mitsubishi Malaysia says it's an SUV. Great, seems the automakers are even more confused.
There was a time not long ago when cars fell into very obvious categories. A sedan had a three box design with a separate boot that extended away from the rear axle whereas a hatchback lacked this to be more compact.
On the other hand, a coupe would typically be defined by two doors and a more svelte profile. A pick-up truck was also clearly defined by its cab-centric shape and cargo bed in the rear. Simple, and these two definitions are still true today, mostly.
Not to bash on Mitsubishi here. Their USA website does, in fact, cite all the correct differences between an SUV and crossover. The issue is that none of those differences still apply. One such contrast here is the vehicle platform and construction. By the traditional definition, SUVs use a body-on-frame design with a separate chassis whereas a crossover has a unibody structure much like a car.
In a time when your typical SUV would be fairly large and designed for durability and maximum modularity, this would be true. In recent times, though, there are only a handful of models that match this description such as the Ford Everest, Isuzu MU-X, Mitsubishi Pajero Sport, Toyota Land Cruiser, Nissan Patrol, Mercedes-Benz G-Class, and Jeep Wrangler.
Ironically, if four-wheel traction and a body-on-frame construction were the only defining conditions for an SUV, then even the Tesla Model 3 would qualify, yet the Range Rover and all-new Land Rover Defender would be categorised as crossovers.
See? It doesn't matter. What matters is what sells.
Much like that American all-electric four-door, however, most SUVs and/or crossovers are just about as useless off-road. Despite being tempted by marketing campaigns/lies promising an 'active lifestyle' of camping and surfing and laughs, automakers are very aware that a huge majority of buyers never leave the safety of a paved road.
Consequently, they've all gone soft. Nearly all these vehicles regardless of their branding campaign as an SUV or crossover are only equipped with front-wheel drive and are fitted with road-only tyres. Your bulky shape and less aerodynamic elevated ride height won't save you either, apart from making your wallet pay for more fuel.
Both crossovers and SUVs do prioritise additional practicality with more space to load cargo, fulfilling the 'Utility' portions of the SUV acronym. That said, they're not appreciably more spacious than a typical 5-door hatch, which is cheaper and yields better fuel economy.
Bottom line, in 2021 both SUVs and crossovers are pretty much interchangeable with any distinction between the two having lost all meaning. The auto industry has successfully tricked us all into buying a more expensive vehicle on a (false) promise which none of their buyers will ever care to verify or hold them accountable to.
Unfortunately, the bar to be a modern-day SUV is set pretty low. They're all Vehicles, of course, but they're not Sporty nor are they big on Utility either. Unless, of course, you're talking about Subarus - they're pretty legit. For the more rugged varieties, calling them off-roaders or 4x4s is probably more accurate.