WatchESPN and ESPN3, which when you type them into a browser resolve to the same website, are functionally different. WatchESPN generally refers the ability to watch the 24/7 networks online or via smartphone. ESPN3 on the other hand is an internet exclusive network and does not have a true TV component to it.
WatchESPN much like ESPN3 is tied to the service provider, but WatchESPN is tied to pay TV subscription, where ESPN3 is tied to your internet service provider.
The big difference when it comes to the two products, right now, is that ESPN3 does not show games from ESPNU, which is why those games only have the WatchESPN logo.
The WatchESPN logo will replace the Mobile ESPN logo put on the site in prior years.