If you have ESPN through a TV provider, you should have access to ESPN content on the mobile app/website/smart TV app, but you won't get access to ESPN+ content. ESPN+ content is considered over-the-top, meaning you have to pay for an additional subscription to get ESPN+ content. With ESPN+ you also get access to their Insider content on the website. Unfortunately, even if you pay for ESPN+, they sometimes have premium content that requires additional pay per view fees, like UFC marquee events.
You can buy ESPN+ without an ESPN TV provider, however you will not get access to programs appearing on the traditional TV channels (ESPN, ESPN2, ESPNU, etc.).
All of the mobile/smart device content is housed in the ESPN app, so no need for separate apps for your phone or smart TV. Your subscription login determines which programs you have access to. Hope this helps.