Ill address GPS first.
It allows you to track your route, many also have an elevation measure so you can track and cross check what your heart rate, cadence and your speed is, this allows you to see where you are lacking and find out where your body is working harder, necessary? No but you can see how this has benefits allowing you to analyze your entire ride and tailor it a little more and pin point it.
If all you know is distance, time and heart rate, all this tells you is you did 50km at a different average heart rate and a different pace every ride. But it doesn't tell you why. This is where having the gps helps, you can say oh my heart rate was higher due to the hills, or it took longer due to the uphill elevation. It really allows you to get into what happening, you can tell if you improved because of fitness or because of route conditions.
Do they improve performance?
If you use them right yes. Just wearing it wont do much if your not training to certain specifics.
They are a tool to give you measures but you need to know what measures you want to see. What they allow you do to is know what heart rate zone you are training in for specific adaptions. They tell you your cadence so you can work on what suits you best and you can improve your ability to control cadence to a optimal level.
You can see where you are hitting the walls too, while you cannot know the power you can see the speed so if on a at home training where the route is obviously constant, you will eventually see either your heart rate rise to maintain speed or your speed drop to maintain heart rate, this is where you're fatiguing. So this is a good measure and you can actually see after training that eventually you will be able to maintain heart rate and speed for longer before fatigue signs kick in.
Overall the best measure is going to be power and threshold but unless you have the money for a power meter a heart rate monitor is a very good tool. using a HRM with a power meter is the ultimate combination.
Conclusion, they are invaluable tools for training and been able to track and be specific with your workouts. Even on the stationary bike they are great, shows heart rate, shows cadence, allows you to induce the adaptions you want by letting you measure and track how you are training.