Do you need App/Play Store distribution, or offline support? If so, you usually need to go native or use something like Cordova to package into a native app. From the GUI building side of things, responsive and Cordova can be considered the same technology, although with some special considerations when running under Cordova (webviews might be a bit different than bleeding edge browsers, native access etc). If you do decide to go responsive/Cordova, do yourself a favour and get to know CSS Flex (great for layout on screens etc). Responsive/Cordova also allow you to use other great stuff like React+Flux.
The one thing that will haunt you if you do not go native is the "feel" of you app, compared to the native apps on the same platforms. Most games solve this problem by having their own "immersive" GUI which often works because GUIs in games are fairly limited. Usually a responsive/Cordova type app will either select one of the native "styles" and mimic that, OR select to mimic a "website" type interaction, but regardless, it will never feel quite native on mobiles/tablets.
From another point of view, if you have to build everything yourself, going native means it's going to be a looong time until you cover all platforms (Android, iOS, responsive web). Depending on your app, doing cross platform first, THEN invest in native when you've got a hit under your belt.