I have this problem on my phone for 2 days now. When I'm connected to the WLAN at home or via mobile data, some apps (Spotify, MyFitnesspal) show that I have no internet connection. I can easily use other apps like Whatsapp and Youtube works without problems.
Google works, but if I want to open websites again comes "No Internet".
I do not have any of these problems on my laptop, so I do not think the problem is with the router.
Simply reset the phone / reset to factory settings.