Far from the groundbreaking experience the commercials make it out to be (you "leave" your car for a series of lame quick-time events), The Run is just a slight twist on the racing game, pitting you in a series of fifty or so "events," in which you'll either be attempting to beat a set timer, pass rivals in a one-on-one race, or get ahead of three to ten other racers. Cops and mafioso might interfere, but the basic gist is like any other drag-racing game -- just with a wider variety of locations, which is NFS's biggest asset here.
For all the quick speed of the racing itself, which has done away with starting flags and engine revving (you always start in the middle of driving), the game itself is pretty slow. The loading is frustrating, especially since the unforgiving -- and largely randomized -- traffic will have you utilizing the limited number of "rewinds" quite frequently. Additionally, changing cars is done on the fly -- which means if you miss a pit stop, you may be stuck with a car that doesn't fit the next leg (and may therefore need to restart the set of races that you're on). Likewise, shortcuts are tricky -- the mini-map highlights them in orange, but if you're not watching it (and it's hard to do so), you may start down paths that are just dead-ends, at which point the game will automatically rewind.
As for the mechanics of the racing itself, the AI still cheats to catch itself up, and the drift mechanism is difficult to engage, especially on opponents who frequently use their nitro to boost out of range. (If you own a copy of the game, the multiplayer may extend the game's life: there are plenty of rewards to unlock through replay.)
To sum up, the engine is there, as is the design, but the frustrating handling slows down the finished product.