How accurately does an exercise bike simulate "real" biking? Are the statistics displayed on the screen realistic?
It's really difficult to say whether the readouts are accurate without reference to the specific readouts and the bike model. However, a few notes:
Different muscles are used
Bicycling primarily targets the glutes, hamstrings, quadriceps, shins and calves. Stationary bicycles put more strain on your hamstrings due to the need to exert muscular effort to slow the flywheel. There's also less variation of muscles used since you're not turning and you're not dealing with going up and down inclines. That said, you're only going to be seeing significant gains there if you're doing heavy activity, which frankly most people aren't on either an indoor or outdoor bike.
It's easier to "coast" with an exercise bike
I think we've all experienced this, getting to a particular speed and the flywheel is pushing your feet almost as much as you're pushing the pedals. Compare that to biking outdoors where, unless you're on a decline, you have to keep putting effort into the system to keep it moving.
Distance traveled and speed tend to be accurate, but calories burned may not be
Speed and distance are essentially a function of rotational speed of the wheel and its diameter. So yes, outside of flywheel coasting, the speed at which you're turning the pedals probably does work out to that speed, and maintaining that speed for a given time will model the distance. Calories are more tricky. First of all, they're generally based on a formula that combines your height, weight, and age, so it's an estimate from the start. Secondly, most exercise machines don't subtract resting metabolic rate, so the figures look slightly more impressive than they are. You burn about 130 calories per hour just sitting at your desk, 180 per hour if standing. A moderate pace on the bike burns about 490 calories per hour. The machines will be indicating the highest figure they can, generally including a chunk of your resting rate. That said, it's not that uncommon for "calories burned" to be listed that way in general, so the machines aren't terribly deceptive there.
If your aim is purely to burn calories, an exercise bike is fine.1
Thought: consider a rowing machine, which is much tougher and burns even more calories.
Regarding the accuracy of calorie calculations: quite simply ALL calorie calculations have a very high margin of error. For instance, if you ride a bike, or indeed go jogging, for X hours, the guess of how many calories burned is very much a guesstimate. Really I'd say the calorie guess on a exercise bike (or a gps watch, or using a formula) is "reasonable".
Regarding the relatively unimportant question of which of an exercise bike versus real biking "burns more calories". At some level of comparison, real biking is more irregular, and hence, burns more calories. (Same with jogging versus treadmill.)
However, exercise bike is a far more consistent exercise experience. If you go out on a real bike (unless you're a real pro, as SeanDuggan suggests) you're messing around a lot: if you sit down on an exercise bike and actually do it for 30 mins, that's 30 real mins of exercise. (Again, consider instead a rowing machine - perhaps alternate each day?)
As long as you stick at it, exercise bike is one of the great basics of aerobic exercise - calorie burning - so sure, go for it.
Your question is quite abrupt so I've attempted to guess at what you're really asking. Cheers
1 (if your aim is to actually "be good and/or fast at bike riding", just forget the exercise bike and ride an actual bike. But I believe this has nothing to do with your question.)