Writing quite a lot of C# at work, the language feels very much like
what I'm used to
That's one point about Dart. Javascript is considered an awkward language with few general idioms. In a language like Java there is often a natural way to approach a problem. For instance if you keep an inventory of table, in Java or C# you will create a class Table.
Javascript has no classes, you might want to use prototypes but they feel awkward and don't provide such strong structure and encapsulation tools. (At least not without making any stunts.) Inheritance, composition etc. is awkward with Javascript prototypes. That's why most people use plain hash maps to store data. Or they use 3rd party libs like prototype which gives you a class-like experience.
So convenience is one thing, structure the other. Javascript just doesn't scale well because there is no standard way to structure large-scale apps. However currently such 3rd party libs are becoming really popular. (Like backbone.js)
Dart is something to solve that. It's there to give you the stuctural convenience of Java and moreover it doesn't have all these awkward JS features. (Most of them related to weak typing.)
So the answer is yes: classes, inheritance, ...: "traditional OOP". (Most realworld JS webapps out there use jQuery's callback based approached as main structure.) And it has a loose form of static typing, that's however not the key selling point.
BTW: you may want to read this "internal" Google mailing dated 2010: Future of Javascript
Javascript has fundamental flaws that cannot be fixed merely by evolving the
language. We'll adopt a two-pronged strategy for the future of Javascript... Develop a new language (called Dash) that aims to maintain the dynamic nature of Javascript but have a better performance profile and be amenable to tooling for large projects...
Is there any advantages of using Dart?
mean? Advantage over what and measured how? There's a range of languages that compile to JavaScript. Most of them do something a lot better than Dart. Some do most things better than Dart. And for example, if you do C#, then ScriptSharp might be interesting for you. While among all the languages compiling to JavaScript it would be far from being my first choice, for you it would present a very easy transition, possibly even allowing to port existent code painlessly.