If you upgrade to a decent microphone, the difference will be noticable. If you use it correctly, the difference will be worth every penny.

The "use it correctly" is the crunch.

Firstly "shotgun" or "short gun" refers to the shape, it looks like a gun barrel and it's a bit misleading in that it can give the impression that it somehow captures sound from a distance. What it does is reject sounds and noises from the sides, so it's more sensitive to sounds from the front, but it won't pick up what you can't hear. In situations like speeches, this rejection of sounds to the sides means that you capture more of what you want to hear, less of what you don't.
This can mean that if you're not paying attention and don't point the microphone at the subject, you don't get what you want. We're not talking sniper accuracy here, as long as it's pointed at the source.

They are also quite sensitive to handling noise and should be used with a rubber mount. Since they also are much more sensitive to the top and bottom ranges of our hearing range, they tend to pick up more wind noise that an on-camera mic.

Edit: When it comes to motor-sports, the sound of the cars will be much better, clearer and with more depth if you use a decent off-camera mic. It's definitely not worth going for cheap though. Anything under about £150 new is dubious and won't be much of an improvement. Once you get into the £300 plus bracket, the improvement is remarkable.

Good names are Rode and Sennheiser, with some of the more expensive Sony mics also worth a look at.