There are two main advantages to accessing your email through the browser: No set up is required and you can access your email from any computer anywhere.
Browser-based email also has it's downsides. If you have many tabs open in your browser, the tab with your email can get lost in the crowd. Sometimes browser-based email "times out" after a certain amount of time and you have to keep logging back in.
The biggest downside to browser-based email, however, is that it isn't integrated in to OS X. There are many places in OS X where you can email directly from the app: email pics from iPhoto, page links from Safari, Tweets, etc. Email sharing is going to be much easier and more prominent in Mountain Lion.
Fortunately it is super easy to set up most browser-based email (Gmail, Yahoo! Mail especially) in OS X's Mail app. When you have your email set up in Mail.app, all the email integration of OS X falls perfectly into place.