There are great websites that tell a story. There are great websites that don’t. Is one better than the other? Not necessarily. Each serves a different purpose. A search engine like google is a great website and an excellent resource. It doesn’t need to tell a story. It needs to be simple and easy to use. It is about determining what kind of company/product/service you are, being comfortable with it and then communicating it to your target audience. If google wasn’t comfortable in their own skin, they may have tried to tell a story, then who knows where they would’ve been. Instead they kept it simple and kept to their purpose.
This not to say I don’t like storytelling websites. I just like them when they fit the purpose. One great storytelling website (which I’ve used before) is Soleil Noir 2012. It has a clearly identified purpose and tells a concise story. An example of storytelling used well in design!