There are a number of factors to consider when choosing a feed reader. Ultimately, you need to assess your working style and pick one that best fits. Some factors worth considering:
Cost: Most desktop and online readers are now free, so cost is not a factor.
Speed: Desktop readers are typically faster than online. Do the little delays of web mail and web browsing make you nuts? Choose a desktop reader.
Features: Feature sets are generally different. Online readers have a wider variety of sharing options than desktop readers. Desktop readers excel at customizability (how often feeds are checked, saving a username and password for checking subscription-only feeds, customizing how long articles are retained after being read, etc.) and handling enhanced subscriptions that include things like video and audio. Have you been following feeds but feel that you want to tweak more settings than you can with your current reader? Choose a desktop reader.
Collaboration and Reuse: Online readers clearly have an edge; notable items are easily shared between friends and you can broadcast and repurpose your favorite articles using a custom RSS feed. Do you want to collaborate heavily with others and create a feed of articles of interest that you can reuse? Choose an online reader.
Multiple Locations: Online readers make working from multiple locations simple but some desktop readers have an online version that you can access when not at your machine (e.g. FeedDemon, NetNewsWire). Some online readers, allow you to access your feed reader, even when you're not online. Do you work at multiple computers? Choose an online reader.
Account Creation and Privacy: Most desktop readers store all data on your computer and it is not shared with anyone. Desktop readers also do not require creation of an account with a third party. (Notable Exception: NewsGator's NetNewsWire and FeedDemon require online account creation and synchronize your data to their online server.) Do you have concerns about creating additional online accounts?
Integration with Reference Managers: Some reference managers have RSS readers built into them. However, there is not an integrated RSS reader that offers near the ease of use or functionality of most standalone RSS readers. There are some ways to transfer data from an RSS manager into a reference manager, but the results are clunky and often unpredictable. Creating your own custom RSS feed from saved or starred items in your RSS reader holds a lot of promise, but doesn't work very well yet.
Web browsers usually use the paradigm of "live bookmarks." Generally, this means that a bookmark is checked for updates regularly and a number often appears after to indicate the number of items of new content. Sometimes a bookmark has a submenu of new content if appropriate.