A fantastic soundtrack is the only thing that has not been improved upon by subsequent games in the series.
NFS3 lived in the days when PC racing games were still reasonably popular. A PC keyboard will always be inferior to analog sticks or a racing wheel when it comes to driving games, but in the late 90s, arcadey series like NFS thrived on the platform. The controls in NFS3 are dead simple: the up arrow key is accelerate, down is brake, and left and right steer the car. The cars have weight to them, but slide easily as well, without needing precise throttle inputs or anything like that that is unattainable with arrow-key controls. However, the controls in NFS3 are very sluggish. The cars aren't terribly responsive, and there is a noticeable delay when you try and reverse your car after hitting a wall, which can easily cost you five to ten seconds.
There are eight tracks and thirteen different cars, and completing certain challenges can unlock one additional track and three new cars. The selection is slim, obviously, but there are represented many of the hot cars of the '90s, such as the Lamborghini Diablo SV and the Chevrolet Corvette C5, plus a few futuristic concept cars. The tracks are worthy of more notice: They are all excellent in design, combining varied landscapes and architecture with shortcuts and multiple routes. There are two tracks for each environment except the unlockable Empire City; all are playable forward, backward, mirrored, and with weather and night effects, creating more variety than is immediately apparent. However, there's plenty of pop-in, and the sluggish cars mean that each lap is very, very long at any track.
The biggest new gameplay mechanic to NFS3 was the return of police, which had made an appearance in the first game but not the second. However, the police gameplay in general is rather primitive. Sure, the police will use spike strips, but anyone who's played Most Wanted or Carbon will remember that those games had heat levels, helicopters, blockades, PIT maneuvers, and so on. You can play as the police, but you will be the only police car against six speeding racers. Both High Stakes and Hot Pursuit 2 have much better "be-the-cop" mechanics.
In fact, Need for Speed High Stakes, the fourth game in the series, is the game you should get if you have any interest in the pre-2000s NFS games. It features all the tracks from NFS3, plus a bunch of new cars and tracks, better controls, a better Hot Pursuit mode, a full career mode, damage modeling, car upgrades, and better graphics.
The one thing that NFS3 has that no NFS game has ever replicated is the soundtrack. It's a fantastic mixture of 6-minute-plus rock and techno loops, all mixed by Rom di Prisco, Saki Kaskas, and Matt Ragan, who, if EA was smart, would write all the Need for Speed soundtracks and presuably save the company a bunch in licensing fees. It's the game's only redeeming factor today...at least in theory. In actuality, certain loops of each song play in certain sectors of each track, which occasionally creates some incredibly intense moments when some particularly fast corners connect with a swell in the music. However, the change between loops is very obvious as you drive along, and you're better off just looking up the songs on Youtube. (User zonda01 has the best versions. [I am not zonda01.])
While Need for Speed III was once the cream of the crop of arcade racers, it has since been overshadowed by none other than its successors, and only its terrific soundtrack still stands out today. If you're looking for a great classic Need for Speed title, look instead to High Stakes or Porsche Unleashed.