Different cross-platform tools have different approaches, but as a concrete example take Corona SDK. You need no native platform knowledge and your app runs inside the Corona runtime, which makes some difficult things easy but some easy things -- third-party SDK integration, install source tracking -- difficult. And if the creator of the cross-platform environment has chosen not to support certain things, you're out of luck (and you're probably always going to be lagging even for things that are supported). On the other hand, you have only a single codebase to maintain and single-click builds for a lot of platforms.
As for Cordova, we spent some time testing if the performance was sufficient when building an early version of Recent News (https://recent.io -- v1.0 just released three days ago!). We concluded it was not. Boot times were slow, the UI felt very non-native, and overall performance didn't meet our requirements. Now that was a while ago and Cordova may have improved, and hardware performance certainly has improved, so maybe it's a better option today.