Triathlon-specific bicycles are generally significantly more aerodynamic than UCI-legal road bicycles. They're specifically designed to be aerodynamic and sacrifice controllability, maneuverability, and comfort all to improve aerodynamics.
And since most everyone else in the race will be riding one, the cut-off time is based on that.
So if you don't ride one, in any serious race you'd be at a huge disadvantage.
How much faster will one make you?
That depends on how strong you are. The stronger you are, the more of a speed advantage a triathlon bike will give you because the absolute improvement in speed will be proportional to your speed without the aerodynamic advantage of a tri bike. 10% faster than "slow" is probably still something like "slow - just not as slow" (I might resemble that....), while 10% faster than "fast" might very well be "screaming fast".
It's hard to quantify the exact improvement you'd see, but using defaults on http://bikecalculator.com/ and setting power to 250 watts, the difference between "hoods" and "aerobars" is 6 kph. At 250W, the default "hoods" data estimates speed to be 34.5 kph. On "aerobars", it's 40.5 kph. Going from 34.5 kph to 40.5 kph is an improvement of 17%. That's huge.
And having ridden a TT bike, that 17% improvement in speed is not an inaccurate value. The actual improvement will depend on specifics (and likely won't be 17%), as others have mentioned. How aerodynamic is your "aero" position? Will you be able to sustain the same power when in that aero position? How well will you be able to climb? But values such as 8% faster or even 12% faster could even be called reasonable.
However, read this: A Case Against TT Bikes
Triathlon bikes (the TT or time trial bikes in the article are nothing more than UCI-legal triathlon bikes) can be dangerous - they've sacrificed everything for aerodynamics and speed.
Will you be able to control that bike at speed? If you're not comfortable hammering away close to your limit for literally a few hours, then the bike might not make you faster at all. It's really hard to ride as hard and as fast as you can when you feel like you're on the verge of losing control and plowing into the nearest tree at 40 kph or crossing the road centerline into oncoming traffic.
So if you do decide you want one, you need to ride it a lot to get comfortable riding it hard for a few hours. And please do not ride your triathlon bike in group rides - they are downright dangerous to everyone in the group. There are good reasons UCI has banned them from mass-start races.
And don't ever be that utter fool "try-athlete" on the local MUP and weave through kids and dogs and crowds at speed while on your aerobars.