I play both....
The Run is definitely more realistic, and Hot Pursuit is like a technical version of Burnout.
The Run is hella unforgiving, the different drivetrains and types of cars really stand out in their given surroundings. Driving in dirt/snow or on a wet road feel nothing like dry tarmac; RWDs will struggle in dirt, AWDs will thrive off it. FWDs and Tuners do really well in city grids, exotic will destroy in technical country sides, muscle cars will kill on freeways, all sound pretty logical, but not something many games addressed or made apparent.
When you're going really fast, your car will skid when you turn, that's natural, and I'm glad that was implemented in the game. When you're traveling to a crest of a hill your car starts lifting off and you lose traction for a bit, if you put one side of the car into a beaten up shoulder lane or dirt, the car will dig itself into it. There's other minute road conditions people don't notice right away, some get annoyed by it (I don't know why), but all that makes driving in this game very tricky. And while it is difficult, the pay off is very satisfying.
Notice how I've only spoken about the driving experience itself and not the story? Forget the story, don't buy the game to play story mode, buy the game if you want a challenging and extremely competitive arcade racer. Wallriding isn't a fully fixed issue, but lots of people who play seriously don't do it.
As for Hot Pursuit 3... don't buy this game if you can't for the life of you understand how drifting works. This game plays like Burnout; you're either in a grip state, or a drift state, there's no braking lines, no grip-cornering lines, nothing traditional. If you've played Burnout before, you should have little to no trouble adapting to this game.
I don't play police chases events anymore, so I only speak from the racing point-of-view.
This game is fast-paced, there's no slowing down at any point. You don't do anything "traditional" to make it through turns, you perform high-speed drifts instead. It's brain-dead, and less technical than The Run. Some people will tell you that this game isn't fun online because you can't catch up when you're falling behind. This is very much untrue. This game has HUGE catch-up boost. Your car won't magically accelerate faster than usual like GT, but you will gain more nitrous than usual. This is not to say that just having more nitrous means you will catch up, you still have to be able to maintain a higher average speed than others to catch up to 1st place.
And if you do get a console version, don't compare your records with those on PC. The mechanics are slightly more favouring for PC players. Regardless of which ever you pick, I am pretty knowledgeable about both, I can link you to another site where these 2 games are the main focus of the place. We've gotten pretty technical about Hot Pursuit so we can give you huge advice on how to get the most of the game.
Final note: At high level play for BOTH games, it's not about who's better at cornering, it's about who can make the best use of nitrous.
It wasn't like that before EA!
