I currently own a Honda Wave 125 with a carburettor.
I'm considering buying a newer fuel injected Honda Wave 110.
What are the power advantages of a fuel injected engine over a carburettor engine?
Would I be better off keeping my 125cc?
I guess the fuel injection may offer features such as wide open throttle fuel enrichment that the carbureted version does not necessarily have. But at the same time you are going to have a 12% reduction in cylinder displacement. I would say that in most cases, the 12% reduction is not offset by the benefits of fuel injection.
This question is impossible to answer with more information. For example, does the fuel injected version have a higher redline RPM than the carbureted version? Also, probably the manufacturer specifies the maximum power for both engines, so you can look at the specifications.
Both fuel injection and carburetors work quite well: they mix a certain amount of fuel into the airflow. Fuel injection is just more precise, and allows more easily adjustable air-fuel ratio which is useful for e.g. cold starts in freezing weather.
The main benefit you get is that you can drive a more environmentally friendly vehicle (less harmful emissions). And you do get somewhat better fuel efficiency as well, meaning you can save some money. But whether the savings offset the higher cost of a fuel injected vehicle is a good question. Usually, it doesn't make sense financially to keep changing vehicles to newer ones.
Personally, I'd choose a fuel injected version. I just hate the exhaust emissions of small vehicles, especially two-stroke ones. Fortunately, both of these vehicles are four-stroke.